1. Introduction
Let
$k\geq 2$
be an integer. For
$N\geq 1$
and a Dirichlet character
$\chi $
mod N, let
$M_k(N,\chi )$
and
$S_k(N,\chi )$
be the space of modular forms and cusp forms of weight k, level N and nebentypus
$\chi $
, respectively. We simply write
$M_k(N)$
and
$S_k(N)$
if
$\chi $
is trivial, and use
$M_k$
and
$S_k$
to denote
$M_k(1)$
and
$S_k(1)$
. For each
$0\leq t\leq k-2$
, the tth period of
$f\in S_k$
is given by
$$ \begin{align} r_t(f):=\int_0^{i\infty}f(z)z^tdz=\frac{t!}{(-2\pi i)^{t+1}}L(f,t+1). \end{align} $$
Here, the L-function of
$f(z)=\sum _{n\geq 1}a_f(n)q^n$
is
$L(f,s)=\sum _{n\geq 1}a_f(n)n^{-s}$
, which converges for
$\operatorname {\mathrm {Re}}(s)$
sufficiently large and can be extended analytically to the whole complex plane. Note that each
$r_t$
can be regarded an element in
$S_k^{\ast }:=\operatorname {\mathrm {Hom}}_{\mathbb {C}}(S_k,\mathbb {C})$
. The result of Eichler and Shimura (see [Reference Eichler6], [Reference Kohnen and Zagier14], [Reference Manin19]) asserts that odd periods
$r_1,r_3,\ldots ,r_{k-3}$
(or even periods
$r_0,r_2,\ldots ,r_{k-2}$
) span the vector space
$S_k^{\ast }$
. However, these periods are not linearly independent. In fact, they are subject to many linear dependence relations, called the Eichler–Shiumra relations [Reference Manin19]. This leads to a natural question. Which periods are linearly independent? The first work in this direction that we are aware of is [Reference Fukuhara7], in which Fukuhara found an explicit subset of odd periods that forms a basis for
$S_k^{\ast }$
. Recently, Lei et al. [Reference Lei, Ni and Xue16], [Reference Lei, Ni and Xue17] have provided some evidence for the linear independence of odd periods and even periods for modular forms, respectively. Furthermore, the linear independence between odd and even periods has been addressed in [Reference Xue26].
In this article, we consider the twisted periods of modular forms, which are natural generalizations of periods. Let
$\chi $
be a primitive Dirichlet character mod D. For
$0\leq n\leq k-2$
, we define the nth twisted period for
$f\in S_k$
[Reference Fukuhara and Yang8, p. 978] to be
$$ \begin{align} r_{n,\chi}(f):=\int_0^{i\infty} f_{\chi}(x) z^n dz=\frac{n!}{(-2\pi i)^{n+1}}L(f,\chi,n+1). \end{align} $$
Here, the twist
$f_{\chi }(z):=\sum _{n\geq 1}\chi (n)a_f(n)q^n$
is an element in
$S_{k}(D^2,\chi ^2)$
[Reference Iwaniec and Kowalski10, Proposition 14.19]. The twisted L-function
$L(f,\chi ,s):=\sum _{n\geq 1}\chi (n)a_f(n)n^{-s}$
, originally defined for
$\operatorname {\mathrm {Re}}(s)\gg 0$
, can be analytically continued to the whole complex plane. For a normalized Hecke eigenform
$f\in S_k$
, we have [Reference Iwaniec and Kowalski10, Proposition 14.20]
$$ \begin{align} \Lambda(f,\chi,s)=i^k w_{\chi} \Lambda(f,\overline{\chi},k-s), \end{align} $$
where
$\Lambda (f,\chi ,s)=(2\pi )^{-s}\Gamma (s)L(f,\chi ,s)$
is the completed twisted L-function and
$ w_{\chi }:=\frac {G(\chi )^2}{D}$
. One can view each
$r_{n,\chi }$
as an element in
$S_k^{\ast }$
.
Since twisted periods can be regarded as elements in
$S_{k}^{\ast }$
, the first question about these periods is whether they span the whole space of
$S_k^{\ast }$
. The case of
$D=4$
has been treated by Kina [Reference Kina12, Corollary 5.8]. The second question is whether we can find Eichler–Shimura-type linear relations among them. Furthermore, we would like to know which periods are linearly independent. The first two questions are beyond the scope of this article. But we will propose related conjectures in the last section. In this article, extending the ideas of [Reference Lei, Ni and Xue16], [Reference Lei, Ni and Xue17], we will provide some evidence for the linear independence of twisted periods.
We now state our results. First, we consider the linear independence of twisted periods for the same character.
Theorem 1.1. Let
$n\geq 1$
be an integer,
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. For
$K\gg _{n,D}1$
, if
$3\leq \ell _1<\ell _2<\cdots <\ell _n\leq \frac {K-2}{2}$
are integers such that
$\chi (-1)=(-1)^{\ell _i}$
for all
$1\leq i\leq n$
, then the set of twisted periods
$\{r_{\ell _i-1,\chi }\}_{i=1}^n$
on
$S_{K}$
is linearly independent.
Theorem 1.2. Let
$n\geq 1$
be an integer,
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. For
$K\gg _{n,D}1$
, if
$3\leq \ell _1<\ell _2<\cdots <\ell _n\leq \frac {K-4}{2}$
are integers such that
$\chi (-1)=(-1)^{\ell _i}$
for all
$1\leq i\leq n$
, then the set of twisted periods
$\{r_{\ell _i,\chi }\}_{i=1}^n$
on
$S_{K}$
is linearly independent.
We then consider the linear independence of twisted periods with the same index but for different characters mod D.
Theorem 1.3. Let
$D\geq 1$
be an odd square-free integer. For
$K\gg _D1$
, if
$3\leq \ell \leq \frac {K-2}{2}$
is an integer,
$\chi _1,\ldots ,\chi _n$
are primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
and
$\chi _i(2)$
are distinct for all
$1\leq i\leq n$
, then the set of twisted periods
$\{r_{\ell -1,\chi _i}\}_{i=1}^n$
on
$S_K$
is linearly independent.
Theorem 1.4. Let
$D\geq 1$
be an odd square-free integer. For
$K\gg _D1$
, if
$3\leq \ell \leq \frac {K-4}{2}$
is an integer and
$\chi _1,\ldots ,\chi _n$
are primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
and
$\chi _2(2)$
are distinct for all
$1\leq i\leq n$
, then the set of twisted periods
$\{r_{\ell ,\chi _i}\}_{i=1}^n$
on
$S_K$
is linearly independent.
Remark 1.1. The assumption that
$\chi _i$
’s take distinct values at
$2$
in Theorems 1.3 and 1.4 can be replaced by
$\chi _i$
’s taking distinct values at an arbitrary fixed prime p, in which case our results hold for
$K\gg _{p,D}1$
.
Remark 1.2. We want to point out that when
$D=1$
, the twisted periods involved in Theorems 1.1 and 1.3 are the odd periods considered in [Reference Lei, Ni and Xue17], and the twisted periods treated in Theorems 1.2 and 1.4 are exactly the even periods studied in [Reference Lei, Ni and Xue16].
Throughout the article, we assume that
$D\geq 1$
is a positive odd square-free integer in order to avoid technical complications.
We sketch the rough idea of the proof of Theorems 1.1 and 1.3. The key is to study the linear independence of the kernel functions that give rise to the twisted periods. For
$f,g\in S_K$
, let
$\langle f,g\rangle $
denote the Petersson inner product. Then, there is a cusp form
$R_{K,n,\chi }$
such that
$r_{n,\chi }(f)=\langle R_{K,n,\chi },f\rangle $
for all
$f\in S_K$
. In general,
$R_{K,n,\chi }$
is expected to have transcendental Fourier coefficient, which is not convenient to study. Instead of considering
$R_{K,n,\chi }$
, we study the modular form
$\operatorname {\mathrm {Tr}}_1^D(G_{\ell ,\chi }(z)G_{K-\ell ,\overline {\chi }}(z))\in M_K$
. Here, the Eisenstein series
$G_{k,\chi }$
of weight k (see Section 2 for details) is given by
$$ \begin{align} G_{k,\chi}(z):&=\sum_{n=0}^{\infty}\sigma_{k-1,\chi}(n)q^n\in M_{k}\left(D,\chi\right) \end{align} $$
$$ \begin{align} \sigma_{k-1,\chi}(n):&=\begin{cases} \frac{(k-1)!D^k}{(-2\pi i)^kG(\overline{\chi})}L(k,\overline{\chi}) &n=0,\\ \sum\limits_{d\mid n}\chi(d)d^{k-1}& n\geq1. \end{cases}\quad \end{align} $$
Additionally, for
$N\mid M$
,
$\operatorname {\mathrm {Tr}}_N^M$
is the trace map
$$ \begin{align} \operatorname{\mathrm{Tr}}_N^M: M_{m}(M)\rightarrow M_m(N),\quad g\mapsto \sum_{\gamma\in\Gamma_0(M)\backslash\Gamma_0(N)}g|_m\gamma, \end{align} $$
where for any real number m and 
the slash operator [Reference Cohen2, Theorem 7.1] is
$$ \begin{align} g(z)|_m\gamma=\det(\gamma)^{m/2} (cz+d)^{-m} g\left(\frac{az+b}{cz+d}\right). \end{align} $$
We will show that the cuspidal projection of
$\operatorname {\mathrm {Tr}}_1^D(G_{\ell ,\chi }(z)G_{K-\ell ,\overline {\chi }}(z))$
is
$$ \begin{align} \mathcal{F}_{K,\ell,\chi}(z):=\operatorname{\mathrm{Tr}}_1^D(G_{\ell,\chi}(z)G_{K-\ell,\overline{\chi}}(z))-\frac{2\sigma_{\ell-1,\chi}(0)\sigma_{k-1,\overline{\chi}}(0)}{\zeta(1-K)}G_K(z), \end{align} $$
where
$G_K(z)=\frac {\zeta (1-K)}{2}+\sum _{n\geq 1}\sigma _{K-1}(n)q^n$
,
$\zeta (1-K)=-\frac {B_K}{K}$
, and
$B_K$
is the Kth Bernoulli number (see Section 3). Then, we can relate the twisted period
$r_{\ell -1,\chi }$
and the cusp form
$\mathcal {F}_{K,\ell ,\chi }$
through the Rankin–Selberg method (Proposition 2.6), which in turn allows us to show the equivalence between the linear independence of
$\{r_{\ell _i-1,\chi }\}_{i=1}^n$
and the linear independence of
$\{\mathcal {F}_{K,\ell _i,\chi }\}_{i=1}^n$
(Proposition 2.9). Applying the same methodology in [Reference Lei, Ni and Xue16], [Reference Lei, Ni and Xue17], we study the linear independence of
$\{\mathcal {F}_{K,\ell _i,\chi }\}_{i=1}^n$
by investigating the matrix
$M_{K,\ell _1,\ldots ,\ell _n,\chi }$
(56) formed by the Fourier coefficients of
$\{\mathcal {F}_{K,\ell _i,\chi }\}_{i=1}^n$
. Similarly, we can show that
$\{r_{\ell -1,\chi _i}\}_{i=1}^n$
is linearly independent if and only if
$\{\mathcal {F}_{K,\ell ,\chi _i}\}_{i=1}^n$
is linearly independent. Theorem 1.3 can be proved by showing that the matrix
$P_{K,\ell , \chi _1,\ldots ,\ell _n}$
(65) formed by the Fourier coefficients of
$\{\mathcal {F}_{K,\ell ,\chi _i}\}_{i=1}^n$
is non-singular.
Theorems 1.2 and 1.4 can be proved in a similar fashion. But we need to consider the Rankin–Cohen bracket (see Section 2) of Eisenstein series. Define
$$ \begin{align}\mathcal{G}_{K,\ell,\chi}(z)=\operatorname{\mathrm{Tr}}^D_1([G_{\ell,\chi}(z),G_{K-2-\ell,\overline{\chi}}(z)]_1). \end{align} $$
By the Rankin–Selberg convolution (Proposition 2.8), we can show the equivalence between the linear independence of
$\{r_{\ell _i,\chi }\}_{i=1}^n$
(
$\{r_{\ell ,\chi _i}\}_{i=1}^n$
) and the linear independence of
$\{\mathcal {G}_{K,\ell _i,\chi }\}_{i=1}^n$
(
$\{\mathcal {G}_{K,\ell ,\chi _i}\}_{i=1}^n$
). Then, Theorems 1.2 and 1.4 can be proved by showing the non-singularity of
$N_{K,\ell _1,\ldots ,\ell _n,\chi }$
(62) and
$Q_{K,\ell , \chi _1,\ldots ,\ell _n}$
(68), respectively.
The article is organized as follows. In Section 2, we prove the equivalence between the linear independence of the twisted periods and the linear independence of the corresponding cusp forms using Rankin–Selberg convolutions. In Section 3, we first compute the Fourier coefficients of
$\mathcal {F}_{K,\ell ,\chi }$
and
$\mathcal {G}_{K,\ell ,\chi }$
. Then, we give the asymptotics of the Fourier coefficients of
$\mathcal {F}_{K,\ell ,\chi }$
and
$\mathcal {G}_{K,\ell ,\chi }$
. In Section 4, we define the matrices
$M_{K,\ell _1,\ldots ,\ell _n,\chi }, N_{K,\ell _1,\ldots ,\ell _n,\chi }, P_{K,\ell ,\chi _1,\ldots ,\chi _n}$
, and
$Q_{K,\ell ,\chi _1,\ldots ,\chi _n}$
formed by the Fourier coefficients of
$\{\mathcal {F}_{K,\ell _i,\chi }\}_{i=1}^n, \{\mathcal {G}_{K,\ell _i,\chi }\}_{i=1}^n, \{\mathcal {F}_{K,\ell ,\chi _i}\}_{i=1}^n$
, and
$\{\mathcal {G}_{K,\ell ,\chi _i}\}_{i=1}^n$
, respectively. We then give criteria for the non-singularity of these matrices. Finally, we prove Theorems 1.1–1.4 by showing the corresponding matrices are non-singular. In Section 5, we give two applications of our method. We prove some identities that evaluate convolution sums of twisted divisor functions. We also show that Maeda’s conjecture implies a non-vanishing result on twisted central L-values of normalized Hecke eigenforms. In the last section, we propose several conjectures on spanning
$S_K$
by twisted periods.
2. Rankin–Selberg convolutions
In this section, we prove the equivalence between the linear independence of twisted periods and the linear independence of the corresponding kernel cusp forms using Rankin–Selberg convolutions.
First, we recall some basic facts of Eisenstein series as developed in Miyake’s book [Reference Miyake20, Section 7]. Let
$\chi $
and
$\psi $
be Dirichlet characters mod L and mod M, respectively. For any positive integer
$k\geq 3$
, we put [Reference Miyake20, Section 7]
$$ \begin{align*} E_{k}(z;\chi,\psi)=\sideset{}{'}\sum\limits_{m,n\in\mathbb{Z}}\chi(m)\psi(n)(mz+n)^{-k}. \end{align*} $$
Here,
$\sideset {}{'}\sum $
is the summation over all pairs of integers
$(m,n)$
except
$(0,0)$
.
Lemma 2.1 [Reference Miyake20, Theorem 7.1.3]
Assume
$k\geq 3$
. Let
$\chi $
and
$\psi $
be Dirichlet characters mod L and mod M, respectively, satisfying
$\chi (-1)\psi (-1)=(-1)^k$
. Let
$m_{\psi }$
be the conductor of
$\psi $
, and
$\psi ^0$
be the primitive character associated with
$\psi $
. Then,
$$ \begin{align*} E_{k}(z;\chi,\psi)=C+A\sum_{n=1}^{\infty}a(n)e^{2\pi inz/M}, \end{align*} $$
where
$$ \begin{align*} A&=2(-2\pi i)^kG(\psi^0)/M^{k}(k-1)!,\nonumber\\ C&=\begin{cases}2L_M(k,\psi) &\chi:\mathrm{the~principal~character},\nonumber\\ 0 &\mathrm{otherwise}, \end{cases}\nonumber\\ a(n)&=\sum_{0<c\mid n}\chi(n/c)c^{k-1}\sum_{0<d\mid(l,c)}d\mu(l/d)\psi^0(l/d)\overline{\psi^0}(c/d).\nonumber \end{align*} $$
Here,
$l=M/m_{\psi }$
,
$\mu $
is the Möbius function,
$L_M(k,\psi )=\sum _{n=1}^{\infty }\psi (n)n^{-k}$
is the Dirichlet series, and
$G(\psi ^0)$
is the Gauss sum of
$\psi ^0$
.
We need the following two types of Eisenstein series in this section. Let
$\chi $
be a primitive Dirichlet character mod D and 
be the principal character.
Define
From Lemma 2, we know that
$$ \begin{align*} G_{k,\chi}(z)&=\sum_{n=0}^{\infty}\sigma_{k-1,\chi}(n)q^n, \nonumber\\ \sigma_{k-1,\chi}(n):&=\begin{cases} \frac{(k-1)!D^k}{(-2\pi i)^kG(\overline{\chi})}L(k,\overline{\chi}) &n=0, \nonumber\\ \sum\limits_{d\mid n}\chi(d)d^{k-1}& n\geq1. \end{cases} \end{align*} $$
The Eisenstein series for the cusp at infinity [Reference Miyake20, p. 272] is defined as
where 
. Note that [Reference Miyake20, p. 273 (7.1.30)]:
The relation between
$G_{k,\chi }(z)$
and
$E_{k,D}^{\ast }(z;\overline {\chi })$
below is needed in the proof of Proposition 2.4.
Lemma 2.2. Let
$\chi $
be a primitive Dirichlet character mod D. Then,
$$ \begin{align} G_{k,\chi}(z)=\frac{(k-1)!D^kL(k,\overline{\chi})}{(-2\pi i)^kG(\overline{\chi})}E^{\ast}_{k,D}(z;\overline{\chi}). \end{align} $$
We then review the classical Rankin–Selberg method. For two elements f and g of
$M_k(N)$
such that
$fg$
is a cuspform, the Petersson inner product is given by
$$ \begin{align} \langle f,g\rangle_{N}=\int_{\Gamma_{0}(N)\backslash\mathbb{H}}f(z)\overline{g(z)}y^kd\mu. \end{align} $$
Here,
$z=x+iy$
and
$d\mu =dxdy/y^2$
is the
$\operatorname {\mathrm {SL}}_2(\mathbb {R})$
-invariant measure on the upper half plane
$\mathbb {H}$
. We use
$\langle \cdot ,\cdot \rangle $
to denote
$\langle \cdot ,\cdot \rangle _{N}$
if the level is clear from the context.
Definition 2.1. Let
$f(z)\in M_{a}(\Gamma )$
and
$g(z)\in M_b(\Gamma )$
be modular forms for some congruence subgroup
$\Gamma $
of weights a and b, respectively. For a nonnegative integer e, we define the eth Rankin–Cohen bracket as
$$ \begin{align} [f(z),g(z)]_e := \sum_{r=0}^e (-1)^r\binom{e+a-1}{e-r}\binom{e+b-1}{r}f(z)^{(r)}g(z)^{(e-r)}, \end{align} $$
where
$f(z)^{(r)}$
is the rth normalized derivative
$f(z)^{(r)}:=\frac {1}{(2\pi i)^r}\frac {d^r f(z)}{dz^r}$
of f. Here,
$a,b$
can be in
$\frac {1}{2}\mathbb {Z}$
and the binomial coefficients are defined through gamma functions. Moreover,
$[f,g]_e\in M_{a+b+2e}(\Gamma )$
and
$[f,g]_e\in S_{a+b+2e}(\Gamma )$
for
$e>1$
(see [Reference Cohen2, Theorem 7.1]). We remark that the Rankin–Cohen bracket defined in Zagier [Reference Zagier27, p. 147 (73)] is related to (14) through
$F_{e}^{(a,b)}(f(z),g(z))= (-2\pi i)^e e![f(z),g(z)]_e$
(see [Reference Lei, Ni and Xue16, p. 122 (1.1)]).
Recall the following classical result on Rankin–Selberg convolutions, which was reformulated and generalized in Zagier [Reference Zagier27], keeping in mind the difference between our definition of the Rankin–Cohen bracket and the one used therein.
Lemma 2.3 [Reference Zagier27, Proposition 6]
Let
$k_1$
and
$k_2$
be real numbers with
$k_2\geq k_1+2>2$
. Let
$f(z)=\sum _{n=1}^{\infty }a(n)q^n$
and
$g(z)=\sum _{n=0}^{\infty }b(n)q^n$
be modular forms in
$S_k(N,\chi )$
and
$M_{k_1}(N,\chi _1)$
, where
$k=k_1+k_2+2e, e\geq 0$
and
$\chi =\chi _1\chi _2$
. Then,
$$ \begin{align} \langle f,[g,E^*_{k_2,N}(\cdot;\chi_2)]_e\rangle_{N}=\frac{(-1)^e}{e!}\frac{\Gamma(k-1)\Gamma(k_2+e)}{(4\pi)^{k-1}\Gamma(k_2)}\sum_{n=1}^{\infty}\frac{a(n)\overline{b(n)}}{n^{k_1+k_2+e-1}}. \end{align} $$
Proposition 2.4. Let
$3\leq \ell \leq \frac {K-2}{2}$
be an integer, let
$\chi $
be a primitive Dirichlet character mod D such that
$\chi (-1)=(-1)^{\ell }$
, and let
$f\in S_K$
be a normalized Hecke eigenform. Then,
$$ \begin{align*} \langle f,G_{\ell,\chi}G_{K-\ell,\overline{\chi}}\rangle_{D}=\frac{\Gamma(K-1)(K-\ell-1)!D^{K-\ell}}{(4\pi)^{K-1}(2\pi i)^{K-\ell}\overline{G(\chi)}}L(f,K-1)L(f,\overline{\chi},K-\ell). \end{align*} $$
Proof. Let
$k=K-\ell $
. Note that (12) implies that
$$ \begin{align} G_{k,\overline{\chi}}(z)=\frac{(k-1)!D^kL(k,\chi)}{(-2\pi i)^{k}G(\chi)}E_{k,D}^{\ast}(z;\chi). \end{align} $$
As
$k\ge \ell +2$
, using Lemma 2.3 for
$e=0$
, (4), and the fact that
$\overline {\sigma _{\ell -1,\chi }(n)}=\sigma _{\ell -1,\overline {\chi }}(n)$
, we get
$$ \begin{align} \langle f,G_{\ell,\chi}G_{K-\ell,\overline{\chi}}\rangle_{D}=\frac{\Gamma(K-1)(K-\ell-1)!D^{K-\ell}\overline{L(K-\ell,\chi)}}{(4\pi)^{K-1}(2\pi i)^{K-\ell}\overline{G(\chi)}}\sum_{n=1}^{\infty}\frac{a_f(n){\sigma_{\ell-1,\overline{\chi}}(n)}}{n^{K-1}}. \end{align} $$
By Lemma 2.5, we have
$$ \begin{align} \sum_{n=1}^{\infty}\frac{a_f(n)\sigma_{\ell-1,\overline{\chi}}(n)}{n^{K-1}}=\frac{L(f,K-1)L(f,\overline{\chi},K-\ell)}{L(K-\ell,\overline{\chi})}. \end{align} $$
Lemma 2.5. Let
$f(z)=\sum _{n\geq 1}a_{f}(n)q^n\in S_K$
be a normalized Hecke eigenform. Then, for a Dirichlet character
$\chi $
, an integer
$\ell \geq 3,$
and a complex number s with
$\operatorname {\mathrm {Re}}(s)>\ell +\frac {K-1}{2}$
, we have
$$ \begin{align*} \sum_{n=1}^{\infty}\frac{a_f(n)\sigma_{\ell-1,\chi}(n)}{n^{s}}=\frac{L(f,s)L(f,\chi,s-\ell+1)}{L(2s-\ell+2-K,\chi)}. \end{align*} $$
Proof. Since
$f\in S_K$
is a normalized Hecke eigenform, we have
$$ \begin{align*} a_f(m)a_f(n)=\sum_{d\mid(m,n)}a_f\left(\frac{mn}{d^2}\right)d^{K-1}. \end{align*} $$
Note that s is within the region of convergence for all the involved series. Using the above Hecke relation, we get
$$ \begin{align*} L(f,s)L(f,\chi,s-\ell+1) &=\sum_{m,n\geq1}a_f(m)a_f(n)\chi(n)(mn)^{-s}n^{\ell-1} \nonumber\\&=\sum\limits_{\substack{m,n\geq1\\d\mid(m,n)}}\chi(n)a_{f}\left(\frac{mn}{d^2}\right)d^{K-1}(mn)^{-s}n^{\ell-1} \nonumber\\&=\sum_{d\geq1}\chi(d)d^{K-1+\ell-1-2s}\sum_{m,n\geq1}\chi(n)a_{f}(mn)(mn)^{-s}n^{\ell-1} \nonumber\\&=L(2s-\ell+2-K,\chi)\sum_{n=1}^{\infty}\frac{a_{f}(n)\sigma_{\ell-1,\chi}(n)}{n^s}, \nonumber \end{align*} $$
where the second to last inequality comes from the change of variables
$m\mapsto dm$
and
$n\mapsto dn$
.
Proposition 2.6. Let
$3\leq \ell \leq \frac {K-2}{2}$
be integers and let
$\chi $
be a primitive Dirichlet character mod D such that
$\chi (-1)=(-1)^{\ell }$
, and let
$f\in S_K$
be a normalized Hecke eigenform. Then,
$$ \begin{align} \langle f,\mathcal{F}_{K,\ell,\chi}\rangle=\frac{\Gamma(K-1)(K-\ell-1)!D^{K-\ell}}{(4\pi)^{K-1}(2\pi i)^{K-\ell}\overline{G(\chi)}}L(f,K-1)L(f,\overline{\chi},K-\ell), \end{align} $$
where
$\mathcal {F}_{K,\ell ,\chi }$
is defined in (8).
Proof. Since
$\langle f, G_{K}\rangle =0$
, and
$\langle f,g\rangle _{M}=\langle f,\operatorname {\mathrm {Tr}}_N^M g\rangle _{N}$
for
$N\mid M$
,
$f\in S_K(N)$
,
$g\in M_{K}(M)$
(see [Reference Gross and Zagier9, p. 271]), we have
$$ \begin{align*} \langle f,\mathcal{F}_{K,\ell,\chi}\rangle_1&=\langle f,\operatorname{\mathrm{Tr}}_1^D(G_{\ell,\chi}(z)G_{K-\ell,\overline{\chi}}(z))\rangle_1 \nonumber\\&= \langle f,G_{\ell,\chi}G_{K-\ell,\overline{\chi}}\rangle_{D}. \nonumber \end{align*} $$
Now, the result follows from Proposition 2.4.
Remark 2.1. We would like to mention that Proposition 2.4 (and thus Proposition 2.6) also holds true for
$\ell =\frac {K}{2}$
(see [Reference Zagier27, p. 146] and [Reference Dickson and Neururer5, Proposition 4.1]).
To tackle Theorems 1.2 and 1.4, we need a similar inner product formula for a normalized Hecke eigenform f and Rankin–Cohen brackets of Eisenstein series.
Proposition 2.7. Let
$3\leq \ell \leq \frac {K-4}{2}$
and
$k=K-2-\ell $
be integers, let
$\chi $
be a primitive Dirichlet character mod D such that
$\chi (-1)=(-1)^{\ell }$
, and let
$f(z)=\sum _{n\geq 1}a_f(n)q^n\in S_K$
be a normalized Hecke eigenform. Then,
$$ \begin{align*} \langle f,[G_{\ell,\chi},G_{k,\overline{\chi}}]_1\rangle_{D}=\frac{-k!(K-2)!D^{k}}{(4\pi)^{K-1}(2\pi i)^{k}\overline{G(\chi)}}L(f,K-2)L(f,\overline{\chi},K-\ell-1). \nonumber \end{align*} $$
Proof. Again, by Lemma 2.3 (for
$k_1=\ell , k_2=k$
), (12), and the fact that
$\overline {\sigma _{\ell -1,\chi }(n)}=\sigma _{\ell -1,\overline {\chi }}(n)$
, we get
$$ \begin{align} \langle f,[G_{\ell,\chi},G_{k,\overline{\chi}}]_1\rangle_{D}&=-\frac{\Gamma(K-1)\Gamma(k+1)}{(4\pi)^{K-1}\Gamma(k)}\cdot\frac{(k-1)!D^k\overline{L(k,\chi)}}{(2\pi i)^{k}\overline{G(\chi)}}\sum_{n=1}^{\infty}\frac{a_f(n)\sigma_{\ell-1,\overline{\chi}}(n)}{n^{K-2}}. \end{align} $$
As
$K-2>\ell +\frac {K-1}{2}$
, we have
$$ \begin{align} \sum_{n=1}^{\infty}\frac{a_f(n)\sigma_{\ell-1,\chi}(n)}{n^{K-2}}&=\frac{L(f,K-2)L(f,\overline{\chi},K-2-\ell+1)}{L(2(K-2)-\ell+2-K,\overline{\chi})} \nonumber\\&=\frac{L(f,K-2)L(f,\overline{\chi},K-\ell-1)}{L(k,\overline{\chi})}, \end{align} $$
The following proposition generalizes [Reference Kohnen and Zagier13, Proposition 1] to the Rankin–Cohen bracket.
Proposition 2.8. Under the hypothesis of Proposition 2.7, we have
$$ \begin{align} \langle f,\mathcal{G}_{K,\ell,\chi}\rangle=\frac{-k!(K-2)!D^{k}}{(4\pi)^{K-1}(2\pi i)^{k}\overline{G(\chi)}}L(f,K-2)L(f,\overline{\chi},K-\ell-1). \end{align} $$
Proof. By definition (9),
$\langle f,\mathcal {G}_{K,\ell ,\chi }\rangle =\langle f,\operatorname {\mathrm {Tr}}_1^D[G_{\ell ,\chi },G_{k,\overline {\chi }}]_1\rangle _1= \langle f,[G_{\ell ,\chi },G_{K-\ell ,\overline {\chi }}]_1\rangle _{D}. $
So the result follows from Proposition 2.7.
We now establish the equivalence between the linear independence of twisted periods and the linear independence of the corresponding cusp forms
$\mathcal {F}_{K,\ell ,\chi }$
or
$\mathcal {G}_{K,\ell ,\chi }$
.
Proposition 2.9. Let
$\chi $
be a primitive Dirichlet character mod D. Let
$n\geq 1$
and
$3\leq \ell _1<\ell _2<\cdots <\ell _n\leq \frac {K-2}{2}$
be integers such that
$\chi (-1)=(-1)^{\ell _i}$
for all
$1\leq i\leq n$
. Then,
$\{\mathcal {F}_{K,\ell _i,\chi }\}_{i=1}^n$
is linearly independent if and only if
$\{r_{\ell _i-1,\chi }\}_{i=1}^n$
is linearly independent.
Proof. Note that Proposition 2.6 together with the definition of twisted period (2) and the functional equation (3) implies that for
$3\leq \ell \leq \frac {K-2}{2}$
and a normalized Hecke eigenform,
$f\in S_K$
$$ \begin{align*} \langle f, \mathcal{F}_{K,\ell,\chi}\rangle=A_{K,\ell,\chi}L(f,K-1) r_{\ell-1,\chi}(f), \nonumber \end{align*} $$
where
$A_{K,\ell ,\chi }$
is some nonzero constant depending only on K,
$\ell ,$
and
$\chi $
, and
$L(f,K-1)\neq 0$
since
$K-1$
is within the region of absolute convergence for
$L(f,s)$
. Let
$\mathcal {H}_K$
denote the set of normalized Hecke eigenforms in
$S_K$
. Then,
$$ \begin{align*} \sum_{i=1}^na_i\mathcal{F}_{K,\ell_i,\chi}=0\quad\mathrm{if~and~only~if}\quad\sum_{i=1}^n\overline{a_i}\langle f_j,\mathcal{F}_{K,\ell_i,\chi}\rangle=0\quad\mathrm{for~all}~f_j\in\mathcal{H}_K. \nonumber \end{align*} $$
Suppose that
$\{\mathcal {F}_{K,\ell _i,\chi }\}_{i=1}^n$
is linearly independent. We claim that
$\{r_{\ell _i-1,\chi }\}_{i=1}^n$
is linearly independent. If
$\sum _{i=1}^n b_ir_{\chi ,\ell _{i}-1}=0\in S_K^{\ast },$
then
$$ \begin{align*} \sum_{i=1}^nb_i r_{\ell_i-1,\chi}(f_j)=L(f_j,K-1)^{-1}\sum_{i=1}^nb_iA_{K,\ell_i,\chi}^{-1}\langle f_j,\mathcal{F}_{K,\ell_i,\chi}\rangle=0~\mathrm{for~all}~f_j\in\mathcal{H}_K, \nonumber \end{align*} $$
which implies that
$\sum _{i=1}^n\overline {b_iA_{K,\ell _i,\chi }^{-1}}\mathcal {F}_{K,\ell _i,\chi }=0\in S_K$
. Since
$\{\mathcal {F}_{K,\ell _i,\chi }\}_{i=1}^n$
is linearly independent, we have
$b_iA_{K,\ell _i,\chi }^{-1}=0$
, implying
$b_i=0$
for all
$1\leq i\leq n$
.
Conversely, suppose that
$\{r_{\ell _i-1,\chi }\}_{i=1}^n$
is linearly independent. We show that
$\{\mathcal {F}_{K,\ell _i,\chi }\}_{i=1}^n$
is linearly independent. If
$\sum _{i=1}^n a_i\mathcal {F}_{K,\ell _i,\chi }=0\in S_K,$
then
$$ \begin{align*}&\sum_{i=1}^n\overline{a_i}\langle f_j,\mathcal{F}_{K,\ell_i,\chi}\rangle=0~\mathrm{for~all}~f_j\in\mathcal{H}_K, \nonumber\end{align*} $$
which implies that
$L(f_j,K-1)\sum _{i=1}^n\overline {a_i}A_{K,\ell _i,\chi }\cdot r_{\ell _i-1,\chi }(f_j)=0~\mathrm { for~all}~f_j\in \mathcal {H}_K$
. Since
$L(f_j,K-1)\neq 0$
for all
$f_j\in \mathcal {H}_K$
and
$\{r_{\ell _i-1,\chi }\}_{i=1}^n$
is linearly independent, we get
$\overline {a_i}A_{K,\ell _i,\chi }=0$
, and thus
$a_i=0$
for all
$1\leq i\leq n$
. This completes the proof.
Proposition 2.10. Let
$\chi $
be a primitive Dirichlet character mod D. Let
$n\geq 1$
and
$3\leq \ell _1<\ell _2<\cdots <\ell _n\leq \frac {K-4}{2}$
be integers such that
$\chi (-1)=(-1)^{\ell _i}$
for all
$1\leq i\leq n$
. Then,
$\{\mathcal {G}_{K,\ell _i,\chi }\}_{i=1}^n$
is linearly independent if and only if
$\{r_{\ell _i,\chi }\}_{i=1}^n$
on
$S_K$
is linearly independent.
Proof. Note that Proposition 2.8 together with the definition of twisted period (2) and the functional equation (3) implies that for
$3\leq \ell \leq \frac {K-4}{2}$
and a normalized Hecke eigenform
$f\in S_K$
, we have
$$ \begin{align*} \langle\mathcal{G}_{K,\ell,\chi},f\rangle=B_{K,\ell,\chi}L(f,K-2) r_{\ell,\chi}(f), \nonumber \end{align*} $$
where
$B_{K,\ell ,\chi }$
is some nonzero constant depending only on K,
$\ell ,$
and
$\chi $
, and
$L(f,K-2)\neq 0$
. Then, the proof proceeds in a similar way as the previous one.
The following two propositions can be proved in the same way. We omit their proofs.
Proposition 2.11. Let
$3\leq \ell \leq \frac {K-2}{2}$
be an integer, and let
$\chi _1,\ldots ,\chi _n$
be primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
and
$\chi _i(2)$
are pairwise distinct for
$1\leq i\leq n$
. Then,
$\{\mathcal {F}_{K,\ell ,\chi _i}\}_{i=1}^n$
is linearly independent if and only if
$\{r_{\ell -1,\chi _i}\}_{i=1}^n$
on
$S_K$
is linearly independent.
Proposition 2.12. Let
$3\leq \ell \leq \frac {K-4}{2}$
be an integer, and let
$\chi _1,\ldots ,\chi _n$
be primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
and
$\chi _i(2)$
are pairwise distinct for
$1\leq i\leq n$
. Then,
$\{\mathcal {G}_{K,\ell ,\chi _i}\}_{i=1}^n$
is linearly independent if and only if
$\{r_{\ell ,\chi _i}\}_{i=1}^n$
on
$S_K$
is linearly independent.
3. Fourier coefficients
In this section, we compute the Fourier coefficients of
$\mathcal {F}_{K,\ell ,\chi }$
(8) and
$\mathcal {G}_{K,\ell ,\chi }$
(9) and study their asymptotic behaviors. First of all, we need an explicit formula for the Eisenstein series
$G_{k,\chi }$
under the action of certain matrices in
$\operatorname {\mathrm {SL}}_2(\mathbb {Z})$
.
3.1. Fourier expansion of Eisenstein series at cusps
We need to introduce another type of Eisenstein series,
$G_{k,\chi _1,\chi _2}(z)$
, defined below. Let
$D=D_1D_2$
be such that
$D_1>0$
. Then, there is a unique decomposition
$\chi =\chi _1\chi _2$
, where
$\chi _1$
and
$\chi _2$
are primitive Dirichlet characters mod
$D_1$
and mod
$D_2$
, respectively. Define
$$ \begin{align} G_{k,\chi_1,\chi_2}(z)&:=\frac{D_1^k(k-1)!}{2(-2\pi i)^k G(\overline{\chi}_{1})}E_{k}(D_1z;\chi_{2},\overline{\chi}_{1}) \end{align} $$
$$ \begin{align} &=\frac{D_1^k(k-1)!}{2(-2\pi i)^k G(\overline{\chi}_{1})}\chi_{2}(D_1)\sideset{}{'}\sum_{\substack{m,n\in\mathbb{Z} \\D_1\mid m}}\frac{\chi_{2}(m)\overline{\chi}_{1}(n)}{(mz+n)^k}. \end{align} $$
We would like to remark that such definition is for the consistency of notation in Section 3.2, although it is not necessarily consistent with those of
$E_k(z;\chi ,\psi )$
. From Lemma 2, we know that
$$ \begin{align*} E_{k}(z;\chi_{2},\overline{\chi}_{1}):&=C+\frac{2(-2\pi i)^kG(\overline{\chi}_{1})}{D_1^k(k-1)!}\sum_{n=1}^{\infty}\left(\sum_{\substack{d_1,d_2>0\\ d_1d_2=n}}\chi_1(d_1)\chi_2(d_2)d_1^{k-1}\right)e^{2\pi inz/D_1}, \nonumber\\C:&=\begin{cases} 2L(k,\overline{\chi}_{1})&D_2=1,\\0&\mathrm{otherwsie}. \end{cases} \nonumber \end{align*} $$
Thus, we have
$$ \begin{align} G_{k,\chi_1,\chi_2}(z)&=\sum_{n\geq0}\sigma_{k-1,\chi_1,\chi_2}(n)q^n, \end{align} $$
$$ \begin{align} \sigma_{k-1,\chi_1,\chi_2}(n)&:=\begin{cases}0&n=0\mathrm{~ and}~D_2\neq1,\\ \frac{(k-1)!D^k}{(-2\pi i)^kG(\overline{\chi})}L(k,\overline{\chi}) &n=0~\mathrm{and}~D_2=1,\\\sum\limits_{\substack{d_1,d_2>0\\ d_1d_2=n}}\chi_1(d_1)\chi_2(d_2)d_1^{k-1} &n\geq1. \end{cases} \end{align} $$
We now compute the Fourier expansion of the Eisenstein series
$G_{k,\chi }(z)$
under the action of certain matrices in
$\operatorname {\mathrm {SL}}_2(\mathbb {Z})$
.
Lemma 3.1. Let
$D=D_1D_2$
with
$D_1>0$
and
$\chi $
be a primitive Dirichlet character mod D such that
$\chi (-1)=(-1)^k$
. If 
is a matrix in
$\operatorname {\mathrm {SL}}_2(\mathbb {Z})$
such that
$\gcd (c,D)=D_1$
then
$$ \begin{align*} \left(G_{k,\chi}\bigg|_k\begin{bmatrix} a & b\\ c& d\end{bmatrix}\right)(z)=\chi_2(c)\chi_1(d)\overline{\chi}_{1}(D_2)\overline{\chi}_{2}(D_1)\frac{G(\overline{\chi}_{1})}{G(\overline{\chi})}G_{k,\chi_1,\overline{\chi}_{2}}\left(\frac{z+c^{\ast}d}{D_2}\right), \nonumber \end{align*} $$
where
$\chi =\chi _1\chi _2$
is the product of primitive characters
$\chi _1$
mod
$D_1$
and
$\chi _2$
mod
$D_2$
, and
$c^{\ast }$
is an integer with
$cc^{\ast }\equiv 1\ \pmod {D_2}$
and
$D_1\mid c^{\ast }$
.
Proof. We follow the idea in [Reference Gross and Zagier9, pp. 273–275]. By equation (10), we have
$$ \begin{align*} \left(\frac{2(-2\pi i)^kG(\overline{\chi})}{(k-1)!D^k}G_{k,\chi}\bigg|_k\begin{bmatrix} a&b\\ c&d\end{bmatrix}\right)(z)&=\sideset{}{'}\sum_{\substack{l,r\in\mathbb{Z}\\D\mid l}}\frac{\overline{\chi}(r)}{(l(az+b)+r(cz+d))^k} \nonumber\\&=\sideset{}{'}\sum_{\substack{l,r\in\mathbb{Z}\\D\mid l}}\frac{\overline{\chi}(r)}{((al+cr)z+bl+dr)^k} \nonumber\\&=\sideset{}{'}\sum_{\substack{m,n\in\mathbb{Z} \\md\equiv nc~\text{mod}~D}}\frac{\overline{\chi}(an-bm)}{(mz+n)^k}, \nonumber \end{align*} $$
where 
and thus
$r=an-bm$
. Since
$md\equiv nc\ \pmod {D}$
, we have
$$ \begin{align} d(an-bm)&=adn-bmd\equiv adn-bcn\equiv n\quad \pmod{D}, \end{align} $$
$$ \begin{align} c(an-bm)&=anc-bcm\equiv adm-bcm\equiv m\quad\pmod{D}. \end{align} $$
Note also that
$\gcd (D_1,D_2)=1$
. Then, (27) and (28) imply that
$$ \begin{align*} \overline{\chi}(an-bm)&=\overline{\chi}_{1}(an-bm)\overline{\chi}_{2}(an-bm) \nonumber\\&=\chi_{1}(d)\overline{\chi}_{1}(n)\chi_{2}(c)\overline{\chi}_{2}(m). \nonumber \end{align*} $$
Since
$D_1,D_2\mid (md-nc)$
,
$(d,D_1)=1, (c,D_2)=1,$
and
$(c,D)=D_1$
, we must have
$D_1\mid m$
; and
$n\equiv c^{\ast }md\ \pmod {D_2}$
. Replacing n by
$n=c^{\ast }md+lD_2$
, and choosing
$c^{\ast }$
to satisfy
$D_1\mid c^{\ast }$
by the Chinese Remainder Theorem, so that
$\overline {\chi }_1(c^{\ast }md+lD_2)=\overline {\chi }_1(l)\overline {\chi }_1(D_2)$
. It follows that
$$ \begin{align} &\left(\frac{2(-2\pi i)^kG(\overline{\chi})}{(k-1)!D^k}G_{k,\chi}\bigg|_k\begin{bmatrix} a&b\\ c&d\end{bmatrix}\right)(z) \nonumber\\&\quad=\sideset{}{'}\sum_{\substack{m,l\in\mathbb{Z}\\ D_1\mid m}}\frac{\chi_{1}(d)\overline{\chi}_{1}(c^{\ast}md+lD_2)\chi_{2}(c)\overline{\chi}_{2}(m)}{(mz+mc^{\ast}d+lD_2)^k} \nonumber\\&\quad=\chi_{2}(c)\chi_{1}(d)\overline{\chi}_{1}(D_2)\sideset{}{'}\sum_{\substack{m,l\in\mathbb{Z}\\ D_1\mid m}}\frac{\overline{\chi}_{2}(m)\overline{\chi}_{1}(l)}{(mz+mc^{\ast}d+lD_2)^k} \nonumber\\&\quad=\chi_{2}(c)\chi_{1}(d)\overline{\chi}_{1}(D_2)D_2^{-k}\sideset{}{'}\sum_{\substack{m,l\in\mathbb{Z}\\ D_1\mid m}}\frac{\overline{\chi}_{2}(m)\overline{\chi}_{1}(l)}{\left(m\frac{z+c^{\ast}d}{D_2}+l\right)^k}. \end{align} $$
Note that (24) implies that
$$ \begin{align} \sideset{}{'}\sum_{\substack{m,l\in\mathbb{Z}\\ D_1\mid m}}\frac{\overline{\chi}_{2}(m)\overline{\chi}_{1}(l)}{\left(m\frac{z+c^{\ast}d}{D_2}+l\right)^k}=\frac{2(-2\pi i)^kG(\overline{\chi}_{1})}{D_1^k(k-1)!}\overline{\chi}_{2}(D_1)G_{k,\chi_1,\overline{\chi}_{2}}\left(\frac{z+c^{\ast}d}{D_2}\right). \end{align} $$
3.2. Computation of the trace
For
$m\geq 1$
and
$f(z)=\sum _{n\geq 0}a_f(n)q^n\in S_{k}(N,\chi ),$
we define the U-operator
$$ \begin{align} U_mf(z)=\frac{1}{m}\sum_{v~\mathrm{mod}~m}f\left(\frac{z+v}{m}\right)=\sum_{n\geq1}a_f(mn)q^n. \end{align} $$
Equivalently, we may write
$$ \begin{align}U_mf(z)=m^{k/2-1}\sum_{v\text{ mod }m}f(z)\bigg|_k\begin{bmatrix} 1 & v\\ 0& m \end{bmatrix}.\end{align} $$
We are ready to compute the Fourier coefficients of
$\mathcal {F}_{K,\ell ,\chi }(z)$
(8).
Proposition 3.2. Let
$3\leq \ell \leq \frac {K-2}{2}$
be integers, and let
$\chi $
be primitive character mod D such that
$\chi (-1)=(-1)^{\ell }$
. Then,
$$ \begin{align*} \operatorname{\mathrm{Tr}}^D_1(G_{\ell,\chi}(z)G_{K-\ell,\overline{\chi}}(z))=\sum_{D=D_1D_2}\overline{\chi}_2(-1)U_{D_2}(G_{\ell,\chi_1,\overline{\chi}_{2}}(z)G_{K-\ell,\overline{\chi}_{1},\chi_2}(z)), \nonumber \end{align*} $$
where the summation is over all decompositions of
$D=D_1D_2$
as a product of two positive integers and
$\chi =\chi _1\chi _2$
is the product of primitive characters
$\chi _1$
mod
$D_1$
and
$\chi _2$
mod
$D_2$
.
Proof. Let
$k=K-\ell $
. We consider the following system of representatives of
$\Gamma _0(D)\backslash \operatorname {\mathrm {SL}}_{2}(\mathbb {Z})$
(see [Reference Gross and Zagier9, p. 276] and [Reference Kayath, Lane, Neifeld, Ni and Xue11, Lemma 3.1]):
$$ \begin{align*} \left\{\begin{bmatrix} 1 & 0 \\ D_1 & 1 \end{bmatrix}\begin{bmatrix} 1 & \mu \\ 0 & 1 \end{bmatrix}~:~ D=D_1D_2,~\mu \text{ mod }D_2\right\}. \end{align*} $$
By Lemma 3.1, we have
$$ \begin{align*} G_{\ell,\chi}(z)\bigg|_{k}\begin{bmatrix} 1 & 0 \\ D_1 & 1 \end{bmatrix}\begin{bmatrix} 1 & \mu \\ 0 & 1 \end{bmatrix} &=\chi_2(D_1)\overline{\chi}_{1}(D_2)\overline{\chi}_{2}(D_1)\frac{G(\overline{\chi}_{1})}{G(\overline{\chi})}G_{\ell,\chi_1,\overline{\chi}_{2}}\left(\frac{z+\mu+D_1^*}{D_2}\right), \nonumber\\ G_{k,\overline{\chi}}(z)\bigg|_{k}\begin{bmatrix} 1 & 0 \\ D_1 & 1 \end{bmatrix}\begin{bmatrix} 1 & \mu \\ 0 & 1 \end{bmatrix} &=\overline{\chi}_{2}(D_1)\chi_1(D_2)\chi_2(D_1)\frac{G(\chi_{1})}{G(\chi)}G_{k,\overline{\chi}_{1},\chi_2}\left(\frac{z+\mu+D_1^*}{D_2}\right), \nonumber \end{align*} $$
where
$D_1^{\ast }D_1\equiv 1\ \pmod {D_2}$
and
$D_1\mid D_1^{\ast }$
. It follows that
$$ \begin{align*}\operatorname{\mathrm{Tr}}_1^D(G_{\ell,\chi}(z)G_{k,\overline{\chi}}(z)) =&\sum_{D_1D_2=D}\sum_{\mu \text{ mod } D_2} G_{\ell,\chi}(z)G_{k,\overline{\chi}}(z)\bigg|_{K}\begin{bmatrix} 1 & 0 \\ D_1 & 1 \end{bmatrix}\begin{bmatrix} 1 & \mu \\ 0 & 1 \end{bmatrix} \nonumber\\ =&\sum_{D_1D_2=D}\sum_{\mu \text{ mod } D_2}G_{\ell,\chi}(z)\bigg|_{\ell}\begin{bmatrix} 1 & 0 \\ D_1 & 1 \end{bmatrix}\begin{bmatrix} 1 & \mu \\ 0 & 1 \end{bmatrix}\cdot G_{k,\overline{\chi}}(z)\bigg|_{k}\begin{bmatrix} 1 & 0 \\ D_1 & 1 \end{bmatrix}\begin{bmatrix} 1 & \mu \\ 0 & 1 \end{bmatrix} \nonumber\\ =&\sum_{D_1D_2=D}\sum_{\mu \text{ mod } D_2}\overline{\chi}_2(-1)D_2^{-1}G_{\ell,\chi_1,\overline{\chi}_{2}}\left(\frac{z+\mu+D_1^*}{D_2}\right)G_{k,\overline{\chi}_{1},\chi_2}\left(\frac{z+\mu+D_1^*}{D_2}\right), \nonumber \end{align*} $$
where we used the fact
$$ \begin{align*}G(\chi_{1})G(\overline{\chi}_{1})=\chi_1(-1)D_1\quad \mathrm{and}\quad G(\chi)G(\overline{\chi})=\chi(-1)D\end{align*} $$
in the last equality (see, e.g., [Reference Cohen3, Corollary 2.1.47 on page 33]). On the other hand,
$$ \begin{align*} U_{D_2}\left(G_{\ell,\chi_1,\overline{\chi}_{2}}(z)G_{k,\overline{\chi}_{1},\chi_2}(z)\right) =&\sum_{v~\text{mod}~D_2}D_2^{K/2-1}G_{\ell,\chi_1,\overline{\chi}_{2}}(z)G_{k,\overline{\chi}_{1},\chi_2}(z)\bigg|_{K}\begin{bmatrix} 1 & v\\ 0 & D_2 \end{bmatrix}\\=&\sum_{v~\text{mod}~D_2}D_2^{K/2-1}G_{\ell,\chi_1,\overline{\chi}_{2}}(z)\bigg|_{\ell}\begin{bmatrix} 1 & v\\ 0 & D_2 \end{bmatrix}\cdot G_{k,\overline{\chi}_{1},\chi_2}(z)\bigg|_k\begin{bmatrix}1 & v\\ 0 & D_2 \end{bmatrix}\\=&\sum_{v~\text{mod}~D_2}D_2^{K/2-1}D_2^{-\ell/2}G_{\ell,\chi_1,\overline{\chi}_{2}}\left(\frac{z+v}{D_2}\right) D_2^{-k/2}G_{k,\overline{\chi}_{1},\chi_2}\left(\frac{z+v}{D_2}\right)\\=&\sum_{v~\text{mod}~D_2}D_2^{-1}G_{\ell,\chi_1,\overline{\chi}_{2}}\left(\frac{z+v}{D_2}\right)G_{k,\overline{\chi}_{1},\chi_2}\left(\frac{z+v}{D_2}\right). \end{align*} $$
It follows that
$$ \begin{align*} & \,\sum_{D=D_1D_2}\overline{\chi}_2(-1)U_{D_2}\left(G_{\ell,\chi_1,\overline{\chi}_{2}}(z)G_{k,\overline{\chi}_{1},\chi_2}(z)\right)\\&\quad=\sum_{D=D_1D_2}\sum_{v~\text{mod}~D_2}\overline{\chi}_2(-1)D_2^{-1}G_{\ell,\chi_1,\overline{\chi}_{2}}\left(\frac{z+v}{D_2}\right)G_{k,\overline{\chi}_{1},\chi_2}\left(\frac{z+v}{D_2}\right), \end{align*} $$
as desired.
Proposition 3.3. Let
$3\leq \ell \leq \frac {K-2}{2}$
be an integer and
$\chi $
be a primitive character mod D such that
$\chi (-1)=(-1)^{\ell }$
. Then, the
$q^n$
-Fourier coefficient of
$\operatorname {\mathrm {Tr}}^D_1(G_{\ell ,\chi }(z)G_{K-\ell ,\overline {\chi }}(z)) $
is
$$ \begin{align} \sum_{D=D_1D_2}\overline{\chi}_2(-1)\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2), \end{align} $$
where
$k=K-\ell $
, the summation is over all decompositions of
$D=D_1D_2$
as a product of two positive integers and
$\chi =\chi _1\chi _2$
is the product of primitive characters
$\chi _1$
mod
$D_1$
and
$\chi _2$
mod
$D_2$
.
Proof. Note that the
$q^n$
-Fourier coefficient of
$U_{D_2}\left (G_{\ell ,\chi _1,\overline {\chi }_{2}}(z)G_{K-\ell ,\overline {\chi }_{1},\chi _2}(z)\right )$
is the
$q^{nD_2}$
-Fourier coefficient of
$G_{\ell ,\chi _1,\overline {\chi }_{2}}(z)G_{K-\ell ,\overline {\chi }_{1},\chi _2}(z)$
, which is
$$ \begin{align*} \sum_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2). \end{align*} $$
Corollary 3.1. Let
$3\leq \ell \leq \frac {K-2}{2}$
be an integer and
$\chi $
be a primitive character mod D such that
$\chi (-1)=(-1)^{\ell }$
. Then,
$\mathcal {F}_{K,\ell ,\chi }$
is a cusp form of weight K and level one.
Proof. Let
$k=K-\ell $
. By Proposition 3.3, the constant term of
$\operatorname {\mathrm {Tr}}^D_1(G_{\ell ,\chi }(z)G_{K-\ell ,\overline {\chi }}(z)) $
is
Recall that the Eisenstein series in
$M_K$
is given by
$$ \begin{align*}G_K(z)=\frac{\zeta(1-K)}{2}+\sum_{n\geq1}\sigma_{K-1}(n)q^n.\end{align*} $$
It follows from its definition (8) that
$$ \begin{align*} \mathcal{F}_{K,\ell,\chi}(z)=\operatorname{\mathrm{Tr}}_1^D(G_{\ell,\chi}(z)G_{K-\ell,\overline{\chi}}(z))-\frac{2\sigma_{\ell-1,\chi}(0)\sigma_{k-1,\overline{\chi}}(0)}{\zeta(1-K)}G_K(z)\in S_K, \end{align*} $$
as desired.
Remark 3.1. Note that Propositions 3.2 and 3.3 and Corollary 3.1 also hold true for
$\ell =\frac {K}{2}$
since the computation of the trace is valid as long as
$3\leq \ell \leq K-3$
. See Kohnen–Zagier [Reference Kohnen and Zagier13, p. 193] for the case when
$\chi $
is quadratic and
$\ell =\frac {K}{2}$
.
We now compute the Fourier coefficients of
$\mathcal {G}_{K,\ell ,\chi }$
(see (9) for the definition).
Proposition 3.4. Let
$3\leq \ell \leq \frac {K-4}{2}$
be an integer and
$\chi $
be a primitive character mod D such that
$\chi (-1)=(-1)^{\ell }$
. Then,
$$ \begin{align*} \operatorname{\mathrm{Tr}}^D_1([G_{\ell,\chi}(z),G_{K-2-\ell,\overline{\chi}}(z)]_1)=\sum_{D=D_1D_2}\overline{\chi}_2(-1)D_2^{-1}U_{D_2}\left([G_{\ell,\chi_1,\overline{\chi}_{2}}(z),G_{K-2-\ell,\overline{\chi}_{1},\chi_2}(z)]_1\right), \end{align*} $$
where the summation is over all decompositions of D as a product of two positive integers and
$\chi =\chi _1\chi _2$
is the product of primitive characters
$\chi _1$
mod
$D_1$
and
$\chi _2$
mod
$D_2$
.
Proof. Let
$k=K-2-\ell $
. Then,
$\operatorname {\mathrm {Tr}}^D_1([G_{\ell ,\chi }(z),G_{K-2-\ell ,\overline {\chi }}(z)]_1)=$
$$ \begin{align*} =&\sum_{D_1D_2=D}\sum_{\mu \text{ mod } D_2} \left[G_{\ell,\chi}(z),G_{k,\overline{\chi}}(z)\right]_1\bigg|_{K}\begin{bmatrix} 1 & 0 \\ D_1 & 1 \end{bmatrix}\begin{bmatrix} 1 & \mu \\ 0 & 1 \end{bmatrix} \\ =&\sum_{D_1D_2=D}\sum_{\mu \text{ mod } D_2}\left[G_{\ell,\chi}(z)\bigg|_{\ell}\begin{bmatrix} 1 & 0 \\ D_1 & 1 \end{bmatrix}\begin{bmatrix} 1 & \mu \\ 0 & 1 \end{bmatrix},G_{k,\overline{\chi}}(z)\bigg|_{k}\begin{bmatrix} 1 & 0 \\ D_1 & 1 \end{bmatrix}\begin{bmatrix} 1 & \mu \\ 0 & 1 \end{bmatrix}\right]_1 \\ =&\sum_{D_1D_2=D}\sum_{\mu \text{ mod } D_2}\overline{\chi}_2(-1)D_2^{-1}\left[G_{\ell,\chi_1,\overline{\chi}_{2}}\left(\frac{z+\mu+D_1^*}{D_2}\right),G_{k,\overline{\chi}_{1},\chi_2}\left(\frac{z+\mu+D_1^*}{D_2}\right)\right]_1, \end{align*} $$
where
$D_1^{\ast }D_1\equiv 1\ \pmod {D_2}$
and
$D_1\mid D_1^{\ast }$
. Note also that
$U_{D_2}\left ([G_{\ell ,\chi _1,\overline {\chi }_{2}}(z),G_{k,\overline {\chi }_{1},\chi _2}(z)]_1\right )= $
$$ \begin{align*} &\sum_{v~\text{mod}~D_2}D_2^{K/2-1}[G_{\ell,\chi_1,\overline{\chi}_{2}}(z),G_{k,\overline{\chi}_{1},\chi_2}(z)]_1\bigg|_{K}\begin{bmatrix} 1 & v\\ 0 & D_2 \end{bmatrix}\\&\quad=\sum_{v~\text{mod}~D_2}D_2^{K/2-1}\left[G_{\ell,\chi_1,\overline{\chi}_{2}}(z)\bigg|_{\ell}\begin{bmatrix} 1 & v\\ 0 & D_2 \end{bmatrix},G_{k,\overline{\chi}_{1},\chi_2}(z)\bigg|_k\begin{bmatrix} 1 & v\\ 0 & D_2 \end{bmatrix}\right]_1\\&\quad=\sum_{v~\text{mod}~D_2}D_2^{K/2-1}\left[D_2^{-\ell/2}G_{\ell,\chi_1,\overline{\chi}_{2}}\left(\frac{z+v}{D_2}\right),D_2^{-k/2}G_{k,\overline{\chi}_{1},\chi_2}\left(\frac{z+v}{D_2}\right)\right]_1\\&\quad=\sum_{v~\text{mod}~D_2}\left[G_{\ell,\chi_1,\overline{\chi}_{2}}\left(\frac{z+v}{D_2}\right),G_{k,\overline{\chi}_{1},\chi_2}\left(\frac{z+v}{D_2}\right)\right]_1. \end{align*} $$
It follows that
$$ \begin{align*} &\sum_{D=D_1D_2}\overline{\chi}_2(-1)D_2^{-1}U_{D_2}\left([G_{\ell,\chi_1,\overline{\chi}_{2}}(z),G_{K-2-\ell,\overline{\chi}_{1},\chi_2}(z)]_1\right) \\&\quad=\sum_{D=D_1D_2}\sum_{v~\text{mod}~D_2}\overline{\chi}_2(-1)D_2^{-1}\left[G_{\ell,\chi_1,\overline{\chi}_{2}}\left(\frac{z+v}{D_2}\right),G_{k,\overline{\chi}_{1},\chi_2}\left(\frac{z+v}{D_2}\right)\right]_1, \end{align*} $$
which gives the result.
Proposition 3.5. Let
$3\leq \ell \leq \frac {K-4}{2}$
be an integer and
$\chi $
be a primitive character mod D such that
$\chi (-1)=(-1)^{\ell }$
. Then, the
$q^n$
-Fourier coefficient of
$ \operatorname {\mathrm {Tr}}^D_1([G_{\ell ,\chi }(z),G_{K-2-\ell ,\overline {\chi }}(z)]_1)$
is
$$ \begin{align} \sum_{D=D_1D_2}\overline{\chi}_2(-1)D_2^{-1}\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)(\ell a_2-k a_1), \end{align} $$
where
$k=K-\ell $
, the summation is over all the decompositions of D as a product of two positive integers, and
$\chi =\chi _1\chi _2$
is the product of primitive characters
$\chi _1$
mod
$D_1$
and
$\chi _2$
mod
$D_2$
.
Proof. By (31), we know that the
$q^n$
-Fourier coefficient of
$U_{D_2}\left ([G_{\ell ,\chi _1,\overline {\chi }_{2}}(z),G_{K-2-\ell ,\overline {\chi }_{1},\chi _2}(z)]_1\right )$
is the
$q^{nD_2}$
-Fourier coefficient of
$[G_{\ell ,\chi _1,\overline {\chi }_{2}}(z),G_{K-2-\ell ,\overline {\chi }_{1},\chi _2}(z)]_1$
. Note that
$$ \begin{align*} G_{\ell,\chi_1,\overline{\chi}_2}^{(r)}(z)=\sum_{n\geq0}n^r\sigma_{\ell-1,\chi_1,\overline{\chi}_2}(n)q^n \quad\mathrm{and}\quad G_{k,\overline{\chi}_1,\chi_2}^{(1-r)}(z)=\sum_{n\geq0}n^{1-r}\sigma_{k-1,\overline{\chi}_1,\chi_2}(n)q^n, \end{align*} $$
which implies that the
$q^{nD_2}$
-Fourier coefficient of
$[G_{\ell ,\chi _1,\overline {\chi }_{2}}(z),G_{K-2-\ell ,\overline {\chi }_{1},\chi _2}(z)]_1$
is
$$ \begin{align*} &\sum_{0\leq r\leq1}(-1)^r\binom{\ell}{1-r}\binom{k}{r}\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}a_1^r\sigma_{\ell-1,\chi_1,\overline{\chi}_2}(a_1)a_2^{1-r}\sigma_{k-1,\overline{\chi}_1,\chi_2}(a_2)\\&\quad=\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)(\ell a_2-k a_1). \end{align*} $$
3.3. Asymptotics
Having obtained the explicit formulas for the Fourier coefficients of
$\mathcal {F}_{K,\ell ,\chi }$
and
$\mathcal {G}_{K,\ell ,\chi }$
, we now investigate their asymptotic behaviors. Let
$a_{K,\ell ,\chi }(n)$
and
$b_{K,\ell ,\chi }(n)$
denote the
$q^n$
- Fourier coefficient of
$\mathcal {F}_{K,\ell ,\chi }(z)$
(8) and
$\mathcal {G}_{K,\ell ,\chi }(z)$
(9), respectively. We first normalize
$\mathcal {F}_{K,\ell ,\chi }(z)$
so that its q-Fourier coefficient becomes
$1$
. Define
$$ \begin{align} \mathfrak{F}_{K,\ell,\chi}(z):=\frac{\mathcal{F}_{K,\ell,\chi}(z)}{a_{K,\ell,\chi}(1)}. \end{align} $$
For
$n\geq 1$
, let
$\mathfrak {a}_{K,\ell ,\chi }(n)$
denote the
$q^n$
-Fourier coefficient of
$\mathfrak {F}_{K,\ell ,\chi }(z)$
. By Proposition 3.3 and Corollary 3.1, we have
$$ \begin{align*} \mathfrak{a}_{K,\ell,\chi}(n)=&\frac{\sum\limits_{D=D_1D_2}\overline{\chi}_2(-1)\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)-\frac{2\sigma_{\ell-1,\chi}(0)\sigma_{k-1,\overline{\chi}}(0)}{\zeta(1-K) }\sigma_{K-1}(n)}{\sum\limits_{D=D_1D_2}\overline{\chi}_2(-1)\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=D_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)-\frac{2\sigma_{\ell-1,\chi}(0)\sigma_{k-1,\overline{\chi}}(0)}{\zeta(1-K) }}, \end{align*} $$
where
$k=K-\ell $
. We rewrite the first term in the numerator. Note that
Dividing both the top and bottom by
$\sigma _{k-1,\overline {\chi }}(0)$
, we get that for
$n\geq 1$
,
$$ \begin{align} \mathfrak{a}_{K,\ell,\chi}(n)=\frac{\sigma_{\ell-1,\chi}(n)+\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\sigma_{k-1,\overline{\chi}}(n)-\frac{2\sigma_{\ell-1,\chi}(0)}{\zeta(1-K) }\sigma_{K-1}(n)+\mathcal{E}_{K,\ell,n,\chi}+\mathscr{E}_{K,\ell,n,\chi}}{1+\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}-\frac{2\sigma_{\ell-1,\chi}(0)}{\zeta(1-K) }+\mathscr{E}_{K,\ell,1,\chi}}, \end{align} $$
where
$$ \begin{align*} \mathcal{E}_{K,\ell,n,\chi}&=\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum_{a_1=1}^{n-1}\sigma_{\ell-1,\chi}(a_1)\sigma_{k-1,\overline{\chi}}(n-a_1), \\ \mathscr{E}_{K,\ell,n,\chi}&=\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum\limits_{\substack{D=D_1D_2\\D_2\neq 1}}\overline{\chi}_2(-1)\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2). \end{align*} $$
We also normalize
$\mathcal {G}_{K,\ell ,\chi }(z)$
such that its q-Fourier coefficient is
$1.$
Define
$$ \begin{align} \mathfrak{G}_{K,\ell,\chi}(z):=\frac{\mathcal{G}_{K,\ell,\chi}(z)}{b_{K,\ell,\chi}(1)}. \end{align} $$
Denote by
$\mathfrak {b}_{K,\ell ,\chi }(n)$
the
$q^n$
-Fourier coefficient of
$\mathfrak {G}_{K,\ell ,\chi }(z)$
. By Proposition 3.5, we have
$$ \begin{align*} b_{K,\ell,\chi}(n)=&\sum_{D=D_1D_2}\overline{\chi}_2(-1)D_2^{-1}\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)(\ell a_2-k a_1)\\=&-kn\sigma_{\ell-1,\chi}(n)\sigma_{k-1,\overline{\chi}}(0)+\ell n\sigma_{k-1,\overline{\chi}}(n)\sigma_{\ell-1,\chi}(0)\\&+\sum_{\substack{ a_1,a_2\geq1 \\a_1+a_2=n}}\sigma_{\ell-1,\chi}(a_1)\sigma_{k-1,\overline{\chi}}(a_2)(\ell a_2-ka_1)\\&+\sum_{\substack{D=D_1D_2\\D_2\neq1}}\overline{\chi}_2(-1)D_2^{-1}\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)(\ell a_2-k a_1). \end{align*} $$
After simplification, we get
$$ \begin{align} \mathfrak{b}_{K,\ell,\chi}(n)= \frac{n\sigma_{\ell-1,\chi}(n)-\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\frac{\ell}{k}n\sigma_{k-1,\overline{\chi}}(n)+\mathcal{R}_{K,\ell,n,\chi}+\mathcal{R}_{K,\ell,n,\chi}^{\prime}+\mathscr{R}_{K,\ell,n,\chi}+\mathscr{R}_{K,\ell,n,\chi}^{\prime}}{1-\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\frac{\ell}{k}+\mathscr{R}_{K,\ell,1,\chi}+\mathscr{R}^{\prime}_{K,\ell,1,\chi}}, \end{align} $$
where
$$ \begin{align*} \mathcal{R}_{K,\ell,n,\chi}&=\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum_{a_1=1}^{n-1}a_1\sigma_{\ell-1,\chi}(a_1)\sigma_{k-1,\overline{\chi}}(n-a_1),\\ \mathcal{R}_{K,\ell,n,\chi}^{\prime}&=-\frac{\ell}{k}\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum_{a_1=1}^{n-1}\sigma_{\ell-1,\chi}(a_1)(n-a_1)\sigma_{k-1,\overline{\chi}}(n-a_1),\\ \mathscr{R}_{K,\ell,n,\chi}&=\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum\limits_{\substack{D=D_1D_2\\D_2\neq 1}}\overline{\chi}_2(-1)D_2^{-1}\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}a_1\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2),\\ \mathscr{R}^{\prime}_{K,\ell,n,\chi}&=-\frac{\ell}{k}\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum\limits_{\substack{D=D_1D_2\\D_2\neq 1}}\overline{\chi}_2(-1)D_2^{-1}\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)a_2\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2). \end{align*} $$
The following inequality will be frequently used later.
Lemma 3.6. For
$n\geq 1,$
we have
$$ \begin{align*} \sqrt{2\pi}n^{n+1/2}e^{-n}<n!<en^{n+1/2}e^{-n}. \end{align*} $$
Proof. The non-asymptotic Stirling’s approximation [Reference Robbins22] claims that for all
$n\geq 1$
, we have
$$ \begin{align*} \sqrt{2\pi n}\left(\frac{n}{e}\right)^ne^{\frac{1}{12n+1}}<n!< \sqrt{2\pi n}\left(\frac{n}{e}\right)^ne^{\frac{1}{12n}}, \end{align*} $$
which implies the result.
We also need the following trivial bounds for
$L(k,\chi )$
for
$k\geq 2$
.
Lemma 3.7. Let
$k\geq 2, N\geq 1$
be integers, and let
$\chi $
be a Dirichlet character mod N. Then,
$$ \begin{align*} 2-\zeta(2)\leq2-\zeta(k)\leq|L(k,\chi)|\leq\zeta(k)\leq\zeta(2). \end{align*} $$
Proof. Note that
$2-\zeta (2)\leq 2-\zeta (k)=1-\sum _{n\geq 2}n^{-k}\leq |L(k,\chi )|\leq \sum _{n\geq 1}n^{-k}=\zeta (k)\leq \zeta (2).$
Lemma 3.8. Let
$k\geq \ell \geq 2$
be integers, let
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. Then,
$$ \begin{align*} \left|\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\right|\leq 6\left(\frac{\ell-1}{k-1}\right)^{\ell-1/2}\left(\frac{2\pi e}{(k-1)D}\right)^{k-\ell}. \end{align*} $$
Proof. By Lemmas 3.6 and 3.7, we have
$$ \begin{align*} \left|\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\right|&= \left|\frac{\frac{(\ell-1)!D^{\ell}}{(-2\pi i)^{\ell}G(\overline{\chi})}L(\ell,\overline{\chi})}{\frac{(k-1)!D^k}{(-2\pi i)^kG(\chi)}L(k,\chi)}\right| \\&\leq\left|\frac{L(\ell,\overline{\chi})}{L(k,\chi)}\right|\frac{(\ell-1)^{\ell-1/2}e\cdot e^{-(\ell-1)}}{(k-1)^{k-1/2}\sqrt{2\pi}e^{-(k-1)}}\left(\frac{2\pi}{D}\right)^{k-\ell}\\&\leq\frac{\zeta(2)}{2-\zeta(2)}\cdot\frac{e}{\sqrt{2\pi}}\left(\frac{\ell-1}{k-1}\right)^{\ell-1/2}\left(\frac{2\pi e}{(k-1)D}\right)^{k-\ell}\\&\leq6\left(\frac{\ell-1}{k-1}\right)^{\ell-1/2}\left(\frac{2\pi e}{(k-1)D}\right)^{k-\ell}, \end{align*} $$
as desired.
In the following several lemmas, we estimate the terms that appear in the expressions of Fourier coefficients (36) and (38).
Lemma 3.9. Let
$3\leq \ell \leq \frac {K-2}{2}$
be an integer, let
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. Then,
$$ \begin{align*} \left|\frac{2\sigma_{\ell-1,\chi}(0)}{\zeta(1-K) }\right|\leq 2\left(\frac{(\ell-1)D}{K-1}\right)^{\ell-1/2}\left(\frac{2\pi e}{K-1}\right)^{K-\ell}. \end{align*} $$
Proof. Again, by Lemmas 3.6 and 3.7, we get
$$ \begin{align} \left|\frac{2\sigma_{\ell-1,\chi}(0)}{\zeta(1-K) }\right|&=\left|\frac{2\frac{(\ell-1)!D^{\ell}}{(-2\pi i)^{\ell}G(\overline{\chi})}L(\ell,\overline{\chi})}{\frac{2(K-1)!\zeta(K)}{(2\pi)^K}}\right| \nonumber\\&\leq\frac{(\ell-1)^{\ell-1/2}e\cdot e^{-(\ell-1)}D^{\ell-1/2}(2\pi)^{K-\ell}\zeta(\ell)}{(K-1)^{K-1/2}\sqrt{2\pi}e^{-(K-1)}\zeta(K)} \nonumber\\&=\frac{e\zeta(\ell)}{\sqrt{2\pi}}\left(\frac{(\ell-1)D}{K-1}\right)^{\ell-1/2}\left(\frac{2\pi e}{K-1}\right)^{K-\ell}\\&\leq\frac{e\zeta(2)}{\sqrt{2\pi}}\left(\frac{(\ell-1)D}{K-1}\right)^{\ell-1/2}\left(\frac{2\pi e}{K-1}\right)^{K-\ell} \nonumber\\&\leq2\left(\frac{(\ell-1)D}{K-1}\right)^{\ell-1/2}\left(\frac{2\pi e}{K-1}\right)^{K-\ell}, \nonumber \end{align} $$
as desired.
Lemma 3.10. Let
$3\leq \ell \leq \frac {K-2}{2}$
and
$n\ge 1$
be integers,
$k=K-\ell $
,
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. For
$\mathcal {E}_{K,\ell ,n,\chi }$
in (36), we have
$$ \begin{align*} |\mathcal{E}_{K,\ell,n,\chi}|\leq9.25\left(\frac{\pi en^2}{(K-2)D}\right)^{\frac{K-1}{2}}. \end{align*} $$
Proof. Note that
$$ \begin{align}\sigma_{\alpha}(n):=\sum_{d\mid n}d^{\alpha}\leq\zeta(\alpha)n^{\alpha}\end{align} $$
for all
$\alpha>1$
and
$n\geq 1$
, see [Reference Nathanson21, p. 245]. Then,
$$ \begin{align*} \left|\sum_{a_1=1}^{n-1}\sigma_{\ell-1,\chi}(a_1)\sigma_{k-1,\overline{\chi}}(n-a_1)\right|&\leq\zeta(\ell-1)\zeta(k-1)\sum_{ a_1=1}^{n-1}a_1^{\ell-1}(n-a_1)^{k-1}\\&\le\zeta(2)^2\sum_{ a_1=1}^{ n-1}a_1^{k-1}(n-a_1)^{k-1}\\&\leq\zeta(2)^2(n-1)\left(\frac{n}{2}\right)^{2k-2}\\&\leq2\zeta(2)^2\left(\frac{n}{2}\right)^{2k-1}. \end{align*} $$
By Lemmas 3.6 and 3.8, we have
$$ \begin{align} \left|\sigma_{k-1,\overline{\chi}}(0)^{-1}\right|&=\frac{(2\pi)^k|G(\chi)|}{(k-1)!D^k|L(k,\chi)|} \nonumber\\ &\leq\frac{(2\pi)^k\sqrt{D}}{\sqrt{2\pi}(k-1)^{k-1/2}e^{-(k-1)}D^k(2-\zeta(k))} \nonumber \\ &=\frac{1}{\sqrt{e}(2-\zeta(k))}\left(\frac{2\pi e}{(k-1)D}\right)^{k-1/2} \end{align} $$
$$ \begin{align}& \leq\frac{1}{\sqrt{e}(2-\zeta(2))}\left(\frac{2\pi e}{(k-1)D}\right)^{k-1/2}. \end{align} $$
It follows that
$$ \begin{align*} \left|\mathcal{E}_{K,\ell,n,\chi}\right|&=\left|\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum_{a_1=1}^{n-1}\sigma_{\ell-1,\chi}(a_1)\sigma_{k-1,\overline{\chi}}(n-a_1)\right|\\ &\leq2\zeta(2)^2\left(\frac{n}{2}\right)^{2k-1}\cdot \frac{1}{\sqrt{e}(2-\zeta(2))}\left(\frac{2\pi e}{(k-1)D}\right)^{k-1/2}\\&=\frac{2\zeta(2)^2}{\sqrt{e}(2-\zeta(2))}\left(\frac{\pi en^2}{2(k-1)D}\right)^{k-1/2}\\&\leq 9.25\left(\frac{\pi en^2}{2(k-1)D}\right)^{k-1/2}, \end{align*} $$
as desired.
We now show that
$\mathcal {R}_{K,\ell ,n,\chi }$
and
$\mathcal {R}_{K,\ell ,n,\chi }^{\prime }$
in (38) have similar types of bounds as
$\mathcal {E}_{K,\ell ,n,\chi }$
.
Lemma 3.11. Let
$3\leq \ell \leq \frac {K-4}{2}$
and
$n\geq 1$
be integers,
$k=K-2-\ell $
,
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. Then,
$$ \begin{align*} |\mathcal{R}_{K,\ell,n,\chi}|\leq9.25\left(\frac{\pi en^2}{(K-2)D}\right)^{\frac{K-1}{2}}. \end{align*} $$
Proof. By (40), we have
$$ \begin{align*} \left|\sum_{a_1=1}^{n-1}a_1\sigma_{\ell-1,\chi}(a_1)\sigma_{k-1,\overline{\chi}}(n-a_1)\right|&\leq\zeta(\ell-1)\zeta(k-1)\sum_{ a_1=1}^{n-1}a_1^{\ell}(n-a_1)^{k-1}\\&\leq\zeta(2)^2\sum_{ a_1=1}^{ n-1}a_1^{k-1}(n-a_1)^{k-1}\\&\leq2\zeta(2)^2\left(\frac{n}{2}\right)^{2k-1}. \end{align*} $$
Using the bound (42), we get
$$ \begin{align*}|\mathcal{R}_{K,\ell,n,\chi}|&=\left|\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum_{a_1=1}^{n-1}a_1\sigma_{\ell-1,\chi}(a_1)\sigma_{k-1,\overline{\chi}}(n-a_1)\right|\\&\leq9.25\left(\frac{\pi en^2}{2(k-1)D}\right)^{k-1/2}, \end{align*} $$
which gives the result.
Lemma 3.12. Let
$3\leq \ell \leq \frac {K-4}{2}$
and
$n\geq 1$
be integers,
$k=K-2-\ell $
,
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. Then,
$$ \begin{align*} |\mathcal{R}^{\prime}_{K,\ell,n,\chi}|\leq 18.5\left(\frac{\pi en^2}{(K-2)D}\right)^{\frac{K-1}{2}}. \end{align*} $$
Proof. Again, by (40), we have
$$ \begin{align*} \left|\sum_{a_1=1}^{n-1}\sigma_{\ell-1,\chi}(a_1)(n-a_1)\sigma_{k-1,\overline{\chi}}(n-a_1)\right|&\leq\zeta(\ell-1)\zeta(k-1)\sum_{a_1=1}^{n-1}a_1^{\ell-1}(n-a_1)\cdot (n-a_1)^{k-1}\\&\leq\zeta(2)^2\sum_{ a_1=1}^{n-1}a_1^{k-3}(n-a_1)^{k-3}(n-a_1)^3\\&\leq\zeta(2)^2\left(\frac{n}{2}\right)^{2k-6}\sum_{a_1=1}^{n-1}(n-a_1)^3. \end{align*} $$
Note that
$$ \begin{align*}\sum_{a_1=1}^{n-1}(n-a_1)^3=\frac{(n-1)^2n^2}{4}.\end{align*} $$
Hence,
$$ \begin{align*} \left|\sum_{a_1=1}^{n-1}\sigma_{\ell-1,\chi}(a_1)(n-a_1)\sigma_{k-1,\overline{\chi}}(n-a_1)\right|&\leq\zeta(2)^2\left(\frac{n}{2}\right)^{2k-6}\frac{n^4}{4}\leq4\zeta(2)^2\left(\frac{n}{2}\right)^{2k-1}. \end{align*} $$
Using the bound (42) again, we get
$$ \begin{align*} |\mathcal{R}_{K,\ell,n,\chi}^{\prime}|&=\left|-\frac{\ell}{k}\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum_{a_1=1}^{n-1}\sigma_{\ell-1,\chi}(a_1)(n-a_1)\sigma_{k-1,\overline{\chi}}(n-a_1)\right|\\&\leq\frac{4\zeta(2)^2}{\sqrt{e}(2-\zeta(2))}\left(\frac{\pi en^2}{2(k-1)D}\right)^{k-1/2} \\&\leq18.5\left(\frac{\pi en^2}{(K-2)D}\right)^{\frac{K-1}{2}}, \end{align*} $$
as desired.
Next, we give the bound of
$\mathscr {E}_{K,\ell ,n,\chi }$
in (36).
Lemma 3.13. Let
$3\leq \ell \leq \frac {K-2}{2}$
and
$n\geq 2$
be integers,
$k=K-\ell $
,
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. Then,
$$ \begin{align*} |\mathscr{E}_{K,\ell,n,\chi}|\leq 16\left(\frac{\pi eDn^2}{K-2}\right)^{\frac{K-1}{2}}. \end{align*} $$
Proof. Note that for all 
(
$D_2\neq 1$
) and
$m\geq 0$
, we have
$|\sigma _{k-1,\chi _1,\chi _2}(m)|\leq \zeta (k-1)m^{k-1}$
by (40). Then,
$$ \begin{align} &\left|\sum_{\substack{D=D_1D_2\\D_2\neq 1 }}\overline{\chi}_2(-1)\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)\right| \nonumber\\ &\leq\zeta(\ell-1)\zeta(k-1)\sum_{\substack{D_2\mid D\\D_2>1}}\sum_{a_1=1}^{nD_2}a_1^{\ell-1}(nD_2-a_1)^{k-1} \nonumber\\ &\leq\zeta(\ell-1)\zeta(k-1)\sum_{\substack{D_2\mid D\\D_2>1}}\sum_{a_1=1}^{nD_2}a_1^{k-1}(nD_2-a_1)^{k-1} \nonumber\\ &\leq\zeta(\ell-1)\zeta(k-1)\sum_{\substack{D_2\mid D}}nD_2\left(\frac{nD_2}{2}\right)^{2k-2} \end{align} $$
$$ \begin{align} &\leq2\zeta(\ell-1)\zeta(k-1)\left(\frac{n}{2}\right)^{2k-1}\sigma_{2k-1}(D) \nonumber\\ &\leq2\zeta(\ell-1)\zeta(k-1)\zeta(K-1)\left(\frac{nD}{2}\right)^{2k-1} \end{align} $$
$$ \begin{align} &\leq2\zeta(2)^3\left(\frac{nD}{2}\right)^{2k-1}. \end{align} $$
Using the bound (42) again, we get
$$ \begin{align} \left|\mathscr{E}_{K,\ell,n,\chi}\right|=&\left|\sigma_{k-1,\overline{\chi}}(0)^{-1}\sum\limits_{\substack{D=D_1D_2\\D_2\neq 1}}\overline{\chi}_2(-1)\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)\right| \nonumber \\ \leq&\frac{1}{\sqrt{e}(2-\zeta(2))}\left(\frac{2\pi e}{(k-1)D}\right)^{k-1/2}2\zeta(2)^3\left(\frac{nD}{2}\right)^{2k-1} \nonumber\\ =&\frac{2\zeta(2)^3}{\sqrt{e}(2-\zeta(2))}\left(\frac{\pi eDn^2}{2(k-1)}\right)^{k-1/2} \nonumber\\ \leq&16\left(\frac{\pi eDn^2}{K-2}\right)^{\frac{K-1}{2}}, \end{align} $$
as desired.
We show that
$\mathscr {R}_{K,\ell ,n,\chi }$
and
$\mathscr {R}^{\prime }_{K,\ell ,n,\chi }$
in (38) have similar bounds.
Lemma 3.14. Let
$3\leq \ell \leq \frac {K-4}{2}$
and
$n\geq 1$
be integers,
$k=K-2-\ell $
,
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. Then,
$$ \begin{align*} |\mathscr{R}_{K,\ell,n,\chi}|\leq 16\left(\frac{\pi eDn^2}{K-2}\right)^{\frac{K-1}{2}}. \end{align*} $$
Proof. Note that
$$ \begin{align*} &\left|\sum\limits_{\substack{D=D_1D_2\\D_2\neq 1}} \overline{\chi}_2(-1)D_2^{-1}\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}a_1\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)\right|\\&\leq\zeta(\ell-1)\zeta(k-1)\sum_{\substack{D_2\mid D\\D_2>1}}\sum_{a_1=1}^{n|D_2|}a_1\cdot a_1^{\ell-1}(nD_2-a_1)^{k-1}\\&\leq\zeta(2)^2\sum_{\substack{D_2\mid D\\D_2>1}}\sum_{a_1=1}^{n|D_2|}a_1^{k-1}(nD_2-a_1)^{k-1}\\ &\leq2\zeta(2)^3\left(\frac{nD}{2}\right)^{2k-1}, \end{align*} $$
as in (45). Thus, similar to (46), we obtain
$$ \begin{align*}|\mathscr{R}_{K,\ell,n,\chi}| \leq16\left(\frac{\pi eDn^2}{K-2}\right)^{\frac{K-1}{2}}, \end{align*} $$
which gives the result.
Lemma 3.15. Let
$3\leq \ell \leq \frac {K-4}{2}$
and
$n\geq 1$
be integers,
$k=K-2-\ell $
,
$D\geq 1$
be an odd square-free integer, and let
$\chi $
be a primitive Dirichlet character mod D. Then,
$$ \begin{align} |\mathscr{R}^{\prime}_{K,\ell,n,\chi}|\leq31\left(\frac{\pi eDn^2}{K-2}\right)^{\frac{K-1}{2}}. \end{align} $$
Proof. Using (40), we see that
$$ \begin{align*} &\left|\sum\limits_{\substack{D=D_1D_2\\D_2\neq 1}}\overline{\chi}_2(-1)D_2^{-1}\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)a_2\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)\right|\\&\leq\zeta(\ell-1)\zeta(k-1)\sum_{\substack{D_2\mid D\\D_2>1}}\sum_{a_1=1}^{n|D_2|}a_1^{\ell-1}(nD_2-a_1)\cdot(nD_2-a_1)^{k-1}\\&\leq\zeta(2)^2\sum_{\substack{D_2\mid D\\D_2>1}}\sum_{a_1=1}^{n|D_2|}a_1^{k-3}(nD_2-a_1)^{k-3}(nD_2-a_1)^3\\&\leq\zeta(2)^2\sum_{\substack{D_2\mid D\\D_2>1}}\left(\frac{nD_2}{2}\right)^{2k-6}\sum_{a_1=1}^{n|D_2|}(nD_2-a_1)^3. \end{align*} $$
Note that
$$ \begin{align*} \sum_{a_1=1}^{n|D_2|}(nD_2-a_1)^3=\frac{(nD_2-1)^2(nD_2)^2}{4}\leq\frac{(nD_2)^4}{4}. \end{align*} $$
Hence,
$$ \begin{align*} &\left|\sum\limits_{\substack{D=D_1D_2\\D_2\neq 1}}\overline{\chi}_2(-1)D_2^{-1}\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)a_2\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)\right|\\&\leq4\zeta(2)^2\sum_{\substack{D_2\mid D\\D_2>1}}\left(\frac{nD_2}{2}\right)^{2k-2}\\&\leq4\zeta(2)^2\left(\frac{n}{2}\right)^{2k-2}\sigma_{2k-2}(D)\\&\leq4\zeta(2)^3\left(\frac{nD}{2}\right)^{2k-1}, \end{align*} $$
which implies that
$$ \begin{align*} |\mathscr{R}^{\prime}_{K,\ell,n,\chi}|&=\left|-\frac{\ell}{k}\cdot\sigma_{k-1,\overline{\chi}}(0)\sum\limits_{\substack{D=D_1D_2\\D_2\neq 1}}\overline{\chi}_2(-1)D_2^{-1}\sum\limits_{\substack{a_1,a_2\geq0\\a_1+a_2=nD_2}}\sigma_{\ell-1,\chi_1,\overline{\chi}_{2}}(a_1)a_2\sigma_{k-1,\overline{\chi}_{1},\chi_2}(a_2)\right|\\&\leq4\zeta(2)^3\left(\frac{nD}{2}\right)^{2k-1}\frac{1}{\sqrt{e}(2-\zeta(2))}\left(\frac{2\pi e}{(k-1)D}\right)^{k-1/2}\\&=\frac{4\zeta(2)^3}{\sqrt{e}(2-\zeta(2))}\left(\frac{\pi eDn^2}{K-2}\right)^{\frac{K-1}{2}}\\&\leq31\left(\frac{\pi eDn^2}{K-2}\right)^{\frac{K-1}{2}}, \end{align*} $$
as desired.
Finally, combining the above results, we give the asymptotics of
$\mathfrak {a}_{K,\ell ,\chi }(n)$
and
$\mathfrak {b}_{K,\ell ,\chi }(n)$
. The case
$n=2^j$
suffices to serve our purpose even though a similar result holds true for general n.
Proposition 3.16. Let
$3\leq \ell \leq \frac {K-2}{2}$
and
$j\geq 0$
be integers. Then,
$$ \begin{align*} \mathfrak{a}_{K,\ell,\chi}(2^j)=\sigma_{\ell-1,\chi}(2^{j})(1+o(1)), \end{align*} $$
where
$o(1)\rightarrow 0$
when K is sufficiently large relative to j and D.
Proof. We know from (36) that
$$ \begin{align} \frac{\mathfrak{a}_{K,\ell,\chi}(2^j)}{\sigma_{\ell-1,\chi}(2^j)}&=\frac{1+\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\frac{\sigma_{k-1,\overline{\chi}}(2^j)}{\sigma_{\ell-1,\chi}(2^j)}-\frac{2\sigma_{\ell-1,\chi}(0)}{\zeta(1-K) }\frac{\sigma_{K-1}(2^j)}{\sigma_{\ell-1,\chi}(2^j)}+\frac{\mathcal{E}_{K,\ell,2^j,\chi}}{\sigma_{\ell-1,\chi}(2^j)}+\frac{\mathscr{E}_{K,\ell,2^j,\chi}}{\sigma_{\ell-1,\chi}(2^j)}}{1+\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}-\frac{2\sigma_{\ell-1,\chi}(0)}{\zeta(1-K) }+\mathscr{E}_{K,\ell,1,\chi}}. \end{align} $$
It suffices to show that the non-constant terms in (48) tend to zero when K is sufficiently large relative to j and D. Note that
$$ \begin{align} |\sigma_{k-1,\overline{\chi}}(2^j)|&=\left|\sum_{n=0}^j\overline{\chi}(2^n)2^{(k-1)n}\right|=\left|\frac{\overline{\chi}(2^{j+1})2^{(k-1)(j+1)}-1}{\overline{\chi}(2)2^{k-1}-1}\right|\leq\frac{2^{(k-1)(j+1)}+1}{2^{k-1}-1}\leq2\cdot 2^{(k-1)j}, \nonumber\\ |\sigma_{\ell-1,\chi}(2^j)|&=\left|\frac{\chi(2^{j+1})2^{(\ell-1)(j+1)}-1}{\chi(2)2^{\ell-1}-1}\right|\geq\frac{2^{(\ell-1)(j+1)}-1}{2^{\ell-1}+1}\geq\frac{1}{2}2^{(\ell-1)j}, \end{align} $$
which implies that
$$ \begin{align} \left|\frac{\sigma_{k-1,\chi}(2^j)}{\sigma_{\ell-1,\overline{\chi}}(2^j)} \right|\leq 4\cdot2^{(k-\ell)j}. \end{align} $$
By Lemma 3.8 and (50), we have
$$ \begin{align} \left|\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\frac{\sigma_{k-1,\overline{\chi}}(2^j)}{\sigma_{\ell-1,\chi}(2^j)}\right| \leq\ & 6\left(\frac{\ell-1}{k-1}\right)^{\ell-1/2}\left(\frac{2\pi e}{(k-1)D}\right)^{k-\ell}4\cdot2^{(k-\ell)j} \nonumber\\ \leq\ &24\left(\frac{\pi e 2^{j+1}}{(k-1)D}\right)^{k-\ell}\\=\ &o(1). \nonumber \end{align} $$
Lemma 3.9 together with (50) (with k replaced by K) gives
$$ \begin{align} \left|\frac{2\sigma_{\ell-1,\chi}(0)}{\zeta(1-K) }\frac{\sigma_{K-1}(2^j)}{\sigma_{\ell-1,\chi}(2^j)}\right|&\leq 2\left(\frac{(\ell-1)D}{K-1}\right)^{\ell-1/2}\left(\frac{2\pi e}{K-1}\right)^{K-\ell}4\cdot 2^{(K-\ell)j} \nonumber\\&=8\left(\frac{(\ell-1)D}{K-1}\right)^{\ell-1/2}\left(\frac{2\pi e\cdot 2^j}{K-1}\right)^{K-\ell} \\&=o(1). \nonumber \end{align} $$
By Lemma 3.10 and (49), we have
$$ \begin{align*} \left|\frac{\mathcal{E}_{K,\ell,2^j,\chi}}{\sigma_{\ell-1,\chi}(2^j)}\right|&\leq 9.25\left(\frac{\pi e2^{2j}}{(K-2)D}\right)^{\frac{K-1}{2}}2\cdot 2^{-(\ell-1)j}=o(1). \end{align*} $$
Lemma 3.13 together with (49) implies that
$$ \begin{align*} \left| \frac{\mathscr{E}_{K,\ell,2^j,\chi}}{\sigma_{\ell-1,\chi}(2^j)}\right|&\leq16\left(\frac{\pi eD4^j}{K-2}\right)^{\frac{K-1}{2}}2\cdot2^{-(\ell-1)j}=o(1). \end{align*} $$
By Lemmas 3.8, 3.9, and 3.13, we have
$$ \begin{align*} \left|\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\right|=o(1),\quad\left|\frac{2\sigma_{\ell-1,\chi}(0)}{\zeta(1-K)}\right|=o(1),\quad\mathrm{ and}\quad|\mathscr{E}_{K,\ell,1,\chi}|=o(1), \end{align*} $$
which completes the proof.
Proposition 3.17. Let
$3\leq \ell \leq \frac {K-4}{2}$
and
$j\geq 0$
be integers. Then,
$$ \begin{align*} \mathfrak{b}_{K,\ell,\chi}(2^j)=2^{j}\sigma_{\ell-1,\chi}(2^{j})(1+o(1)), \end{align*} $$
where
$o(1)\rightarrow 0$
when K is sufficiently large relative to j and D.
Proof. We know from (38) that
$ \frac {\mathfrak {b}_{K,\ell ,\chi }(2^j)}{2^{j}\sigma _{\ell -1,\chi }(2^{j})}=$
$$ \begin{align} \frac{1-\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\frac{\ell}{k}\frac{\sigma_{k-1,\overline{\chi}}(2^j)}{\sigma_{\ell-1,\chi}(2^{j})}+\frac{\mathcal{R}_{K,\ell,2^j,\chi}}{2^{j}\sigma_{\ell-1,\chi}(2^{j})}+\frac{\mathcal{R}_{K,\ell,2^j,\chi}^{\prime}}{2^j\sigma_{\ell-1,\chi}(2^{j})}+\frac{\mathscr{R}_{K,\ell,2^j,\chi}}{2^j\sigma_{\ell-1,\chi}(2^{j})}+\frac{\mathscr{R}_{K,\ell,2^j,\chi}^{\prime}}{2^j\sigma_{\ell-1,\chi}(2^{j})}}{1-\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\frac{\ell}{k}+\mathscr{R}_{K,\ell,1,\chi}+\mathscr{R}^{\prime}_{K,\ell,1,\chi}}. \end{align} $$
It suffices to show that the non-constant terms in (53) tend to zero when K is sufficiently large relative to j and D. Note that (51) implies that
$$ \begin{align*} \left|\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\frac{\ell}{k}\frac{\sigma_{k-1,\overline{\chi}}(2^j)}{\sigma_{\ell-1,\chi}(2^j)}\right|=o(1). \end{align*} $$
$$ \begin{align} \left|\frac{\mathcal{R}_{K,\ell,2^j,\chi}}{2^{j}\sigma_{\ell-1,\chi}(2^{j})}\right|&\leq 9.25\left(\frac{\pi e4^j}{(K-2)D}\right)^{\frac{K-1}{2}}2^{-j}\cdot 2\cdot2^{-(\ell-1)j}=o(1). \end{align} $$
$$ \begin{align*} \left|\frac{\mathcal{R}^{\prime}_{K,\ell,2^j,\chi}}{2^{j}\sigma_{\ell-1,\chi}(2^{j})}\right|\leq18.5\left(\frac{\pi e4^j}{(K-2)D}\right)^{\frac{K-1}{2}}2^{-j}\cdot 2\cdot2^{-(\ell-1)j}=o(1). \end{align*} $$
By Lemma 3.14 and (49), we get
$$ \begin{align*} \left| \frac{\mathscr{R}_{K,\ell,2^j,\chi}}{2^j\sigma_{\ell-1,\chi}(2^{j})}\right|&\leq16\left(\frac{\pi eD4^j}{K-2}\right)^{\frac{K-1}{2}}2^{-j}\cdot 2\cdot 2^{-(\ell-1)j}=o(1). \end{align*} $$
By Lemma 3.15 and (49), we have
$$ \begin{align*} \left| \frac{\mathscr{R}_{K,\ell,2^j,\chi}^{\prime}}{2^j\sigma_{\ell-1,\chi}(2^{j})}\right|&\leq31\left(\frac{\pi eD4^j}{K-2}\right)^{\frac{K-1}{2}}2^{-j}\cdot 2\cdot 2^{-(\ell-1)j}=o(1). \end{align*} $$
By Lemmas 3.8, 3.14, and 3.15, we get
$$ \begin{align} \left|\frac{\sigma_{\ell-1,\chi}(0)}{\sigma_{k-1,\overline{\chi}}(0)}\frac{\ell}{k}\right|=o(1),\quad\left|\mathscr{R}_{K,\ell,1,\chi}\right|=o(1),\quad\mathrm{ and}\quad\left|\mathscr{R}^{\prime}_{K,\ell,1,\chi}\right|=o(1). \end{align} $$
This completes the proof.
4. Coefficient matrices
In this section, we define matrices that are used to study the linear independence of the corresponding twisted periods. More precisely, we prove Theorems 1.1–1.4 by showing the matrices, formed by the (normalized)
$q^{2^i}$
-Fourier coefficients of relevant kernel functions for
$0\le i\le n-1$
, are non-singular.
We start with the proof of Theorem 1.1. For
$1\leq i\leq n$
, we pick out the Fourier coefficients
$\mathfrak {a}_{K,\ell _i,\chi }(1)$
,
$\mathfrak {a}_{K,\ell _i,\chi }(2)$
,
$\mathfrak {a}_{K,\ell _i,\chi }(2^2)$
,
$\ldots $
,
$\mathfrak {a}_{K,\ell _i,\chi }(2^{n-1})$
of each
$\mathfrak {F}_{K,\ell _i,\chi }$
(35) and place them in an
$(n,n)$
-matrix
$$ \begin{align} M_{K,\ell_1,\ldots,\ell_n,\chi}:=\left[a_{K,\ell_i,\chi}(2^{j-1})\right]_{1\leq i,j\leq n}=\left[\sigma_{\ell_i-1,\chi}(2^{j-1})(1+\epsilon_{K,i,j})\right]_{1\leq i,j\leq n}. \end{align} $$
To show that
$\{\mathfrak {F}_{K,\ell _i,\chi }\}_{i=1}^n$
is linearly independent, it suffices to show that
$M_{K,\ell _1,\ldots ,\ell _n}$
is non-singular. Such a choice of
$M_{K,\ell _1,\ldots ,\ell _n,\chi }$
allows us to relate it to the well-known Vandermonde matrix in terms of determinant. More specifically, we ignore the error terms
$\epsilon _{K,i,j}$
(
$1\leq i,j\leq n$
) in
$M_{K,\ell _1,\ldots ,\ell _n,\chi }$
and define the following matrix:
$$ \begin{align} \mathbf{M}_{K,\ell_1,\ldots,\ell_n,\chi}:&=\begin{bmatrix} 1 & \sigma_{\ell_1-1,\chi}(2) & \sigma_{\ell_1-1,\chi}(2^2) & \cdots & \sigma_{\ell_1-1,\chi}(2^{n-1})\\ 1 & \sigma_{\ell_2-1,\chi}(2) & \sigma_{\ell_2-1,\chi}(2^2) & \cdots & \sigma_{\ell_2-1,\chi}(2^{n-1})\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & \sigma_{\ell_{n-1}-1,\chi}(2) & \sigma_{\ell_{n-1}-1,\chi}(2^2) & \cdots & \sigma_{\ell_{n-1}-1,\chi}(2^{n-1})\\ 1 & \sigma_{\ell_n-1,\chi}(2) & \sigma_{\ell_n-1,\chi}(2^2) & \cdots & \sigma_{\ell_n-1,\chi}(2^{n-1}) \end{bmatrix}. \end{align} $$
We have the following lower bound for
$\det \mathbf {M}_{K,\ell _1,\ldots ,\ell _n,\chi }$
.
Lemma 4.1. Let
$\mathbf {M}_{K,\ell _1,\ldots ,\ell _n,\chi }$
be defined as in (57). Then,
$$ \begin{align*} \left|\det \mathbf{M}_{K,\ell_1,\ldots,\ell_n,\chi}\right|\geq\left(\frac{3}{4}\right)^{\frac{n(n-1)}{2}}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}. \end{align*} $$
Proof. After column reductions, we have
$$ \begin{align*} \left|\det \mathbf{M}_{K,\ell_1,\ldots,\ell_n,\chi}\right|&=\left|\det\begin{bmatrix} 1 & \chi(2)2^{\ell_1-1} & \cdots & (\chi(2)2^{\ell_1-1})^{n-1}\\ \vdots & \vdots & \ddots & \vdots\\ 1 & \chi(2)2^{\ell_{n}-1} & \cdots & (\chi(2)2^{\ell_{n}-1})^{n-1} \end{bmatrix}\right|\\&=\left|\prod_{1\leq i<j\leq n}\left(\chi(2)2^{\ell_j-1}-\chi(2)2^{\ell_i-1}\right)\right|\\&=\prod_{1\leq i<j\leq n}2^{\ell_j-1}\cdot(1-2^{\ell_i-\ell_j})\\&\geq\prod_{1\leq i<j\leq n}\frac{3}{4}\cdot2^{\ell_j-1}\\&=\left(\frac{3}{4}\right)^{\frac{n(n-1)}{2}}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}, \end{align*} $$
where the second equality is from the determinant of the Vandermonde matrix.
Lemma 4.2. For any
$\tau \in S_n$
, we have
$$ \begin{align*} \left|\prod_{i=1}^n\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1})\right|<2^{n-1}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}. \end{align*} $$
Proof. Note that
$$ \begin{align*} \left|\prod_{i=1}^n\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1})\right|\leq\prod_{i=1}^n\sigma_{\ell_i-1}(2^{\tau(i)-1}). \end{align*} $$
By [Reference Lei, Ni and Xue17, Corollary 2.5], we know that
$\prod _{i=1}^n\sigma _{\ell _i-1}(2^{\tau (i)-1})$
attains maximum when
$\tau $
is the identity permutation. It follows that
$$ \begin{align} \left|\prod_{i=1}^n\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1})\right|\leq\prod_{i=1}^n\sigma_{\ell_i-1}(2^{i-1})<\prod_{i=1}^n2\cdot2^{(i-1)(\ell_i-1)}=2^{n-1}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}, \end{align} $$
which finishes the proof.
We now give a criterion for the non-singularity of
$M_{K,\ell _1,\ldots ,\ell _n,\chi }$
(see also [Reference Lei, Ni and Xue17, Lemma 2.7]).
Proposition 4.3. Define a function
$$ \begin{align} f(x_1,\ldots,x_n):=\left(\prod_{i=1}^n(1+x_i)\right)-1. \end{align} $$
Suppose that
$$ \begin{align} \sup_{\tau\in S_n}|f(\epsilon_{K,1,\tau(1)},\dots,\epsilon_{K,n,\tau(n)})|<n!^{-1}\cdot\left(\frac{3}{4}\right)^{\frac{n(n-1)}{2}}\cdot2^{1-n}. \end{align} $$
Then,
$M_{K,\ell _1,\ldots ,\ell _n,\chi }$
is non-singular.
Proof. Note that
$$ \begin{align*} \det M_{K,\ell_1,\ldots,\ell_n,\chi}&=\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}\prod_{i=1}^n\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1})(1+\epsilon_{K,i,\tau(i)})\\&=\det \mathbf{M}_{K,\ell_1,\ldots,\ell_n,\chi}+\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}f(\epsilon_{K,1,\tau(1)},\ldots,\epsilon_{K,n,\tau(n)})\prod_{i=1}^n\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1}). \end{align*} $$
On the one hand, we have from Lemma 4.1 that
$$ \begin{align*} \left|\det \mathbf{M}_{K,\ell_1,\ldots,\ell_n,\chi}\right|\geq\left(\frac{3}{4}\right)^{\frac{n(n-1)}{2}}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}. \end{align*} $$
On the other hand, we have
$$ \begin{align*} &\left|\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}f(\epsilon_{K,1,\tau(1)},\ldots,\epsilon_{K,n,\tau(n)})\prod_{i=1}^n\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1})\right|\\&\leq n! \sup_{\tau\in S_n}|f(\epsilon_{K,1,\tau(1)},\ldots,\epsilon_{K,n,\tau(n)})|\left|\prod_{i=1}^n\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1}) \right|\\ &<\left(\frac{3}{4}\right)^{\frac{n(n-1)}{2}}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}. \end{align*} $$
Therefore,
$\det M_{K,\ell _1,\ldots ,\ell _n}\neq 0$
.
Lemma 4.4. Let f be defined as in (59). Suppose there is an
$M>0$
such that
$|x_i|\leq M$
for all
$1\leq i\leq n$
. Then,
$$ \begin{align*} |f(x_1,\ldots,x_n)|\leq(1+M)^{n}-1. \end{align*} $$
Proof. Note that
$$ \begin{align} |f(x_1,\ldots,x_n)|&=\left|\prod_{1\leq i\leq n}(1+x_i)-1\right| \nonumber\\&= \left|\sum_{i=1}^nx_{i}+\sum_{1\leq i<j\leq n}x_ix_j+\cdots+x_1x_2\dots x_n\right| \nonumber\\&\leq\binom{n}{1}M+\binom{n}{2}M^2+\cdots+\binom{n}{n}M^n \nonumber\\&=(1+M)^n-1,\end{align} $$
as desired.
Theorem 1.1 follows from the proposition below.
Proposition 4.5. Let
$n\geq 1$
be an integer,
$D\geq 1$
be an odd square-free integer, and
$\chi $
be a primitive Dirichlet character mod D. For
$K\gg _{n,D}1$
, if
$3\leq \ell _1<\ell _2<\cdots <\ell _n\leq \frac {K-2}{2}$
are integers such that
$\chi (-1)=(-1)^{\ell _i}$
for all
$1\leq i\leq n$
, then
$M_{K,\ell _1,\ldots ,\ell _n,\chi }$
is non-singular.
Proof. From Proposition 3.16, we know that
$\epsilon _{K,i,\tau (i)}=o(1)$
for all
$1\leq i\leq n$
and
$\tau \in S_n$
, where
$o(1)\rightarrow 0$
when K is sufficiently large relative to n and D. Then, Lemma 4.4 implies that
$$ \begin{align*} \sup_{\tau\in S_n}|f(\epsilon_{K,1,\tau(1)},\dots,\epsilon_{K,n,\tau(n)})|&\leq(1+o(1))^n-1=o(1). \end{align*} $$
In particular, K can be chosen such that (60) is satisfied, which shows that
$M_{K,\ell _1,\ldots ,\ell _n,\chi }$
is non-singular for
$K\gg _{n,D}1$
.
Remark 4.1. It is possible to obtain an explicit lower bound of K in terms of n and D for Proposition 4.5. However, it will be too messy. Thus, we omit it.
We now consider the matrix that is used to prove Theorem 1.2. For
$1\leq i\leq n,$
we pick out the Fourier coefficients
$\mathfrak {b}_{K,\ell _i,\chi }(1), \mathfrak {b}_{K,\ell _i,\chi }(2), \mathfrak {b}_{K,\ell _i,\chi }(2^2),\ldots ,\mathfrak {b}_{K,\ell _i,\chi }(2^{n-1})$
of each
$\mathfrak {G}_{K,\ell _i,\chi }$
(37) and place them in an
$(n,n)$
-matrix
$$ \begin{align} N_{K,\ell_1,\ldots,\ell_n,\chi}:=\left[\mathfrak{b}_{K,\ell_i,\chi}(2^{j-1})\right]_{1\leq i,j\leq n}=\left[2^{j-1}\sigma_{\ell_i-1,\chi}(2^{j-1})(1+\delta_{K,i,j})\right]_{1\leq i,j\leq n}. \end{align} $$
To prove that
$\{\mathfrak {G}_{K,\ell _i,\chi }\}_{i=1}^n$
is linearly independent, it suffices to show that
$N_{K,\ell _1,\ldots ,\ell _n,\chi }$
is non-singular. Define the matrix
$$ \begin{align} \mathbf{N}_{K,\ell_1,\ldots,\ell_n,\chi}:&=\begin{bmatrix} 1 & 2\sigma_{\ell_1-1,\chi}(2) & 2^2\sigma_{\ell_1-1,\chi}(2^2) & \cdots & 2^{n-1}\sigma_{\ell_1-1,\chi}(2^{n-1})\\ 1 & 2\sigma_{\ell_2-1,\chi}(2) & 2^2\sigma_{\ell_2-1,\chi}(2^2) & \cdots & 2^{n-1}\sigma_{\ell_2-1,\chi}(2^{n-1})\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 2\sigma_{\ell_{n-1}-1,\chi}(2) & 2^2\sigma_{\ell_{n-1}-1,\chi}(2^2) & \cdots & 2^{n-1}\sigma_{\ell_{n-1}-1,\chi}(2^{n-1})\\ 1 & 2\sigma_{\ell_n-1,\chi}(2) & 2^2\sigma_{\ell_n-1,\chi}(2^2) & \cdots & 2^{n-1}\sigma_{\ell_n-1,\chi}(2^{n-1}) \end{bmatrix}. \end{align} $$
Lemma 4.6. Let
$\mathbf {N}_{K,\ell _1,\ldots ,\ell _n,\chi }$
be defined as in (63). Then,
$$ \begin{align*} \left|\det \mathbf{N}_{K,\ell_1,\ldots,\ell_n,\chi}\right|\geq\left(\frac{3}{2}\right)^{\frac{n(n-1)}{2}}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}. \end{align*} $$
Proof. Since
$ \det \mathbf {N}_{K,\ell _1,\ldots ,\ell _n,\chi }=2^{\frac {n(n-1)}{2}}\det \mathbf {M}_{K,\ell _1,\ldots ,\ell _n,\chi }, $
the result follows from Lemma 4.1.
Lemma 4.7. Let
$\tau \in S_n$
be a permutation of the set
$\{1,2,\ldots ,n\}$
. Then,
$$ \begin{align*} \prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell_i-1}(2^{\tau(i)-1}) \end{align*} $$
reaches its maximum exactly when
$\tau $
is the identity permutation.
Proof. Note that [Reference Lei, Ni and Xue17, Lemma 2.4] asserts that for
$\ell _2>\ell _1\geq 2$
and
$n_2>n_1\geq 0$
, we have
$$ \begin{align*} \sigma_{\ell_2-1}(2^{n_2})\cdot\sigma_{\ell_1-1}(2^{n_1})\geq \sigma_{\ell_2-1}(2^{n_1})\cdot\sigma_{\ell_1-1}(2^{n_2}), \end{align*} $$
which implies that
$$ \begin{align*} 2^{n_2}\sigma_{\ell_2-1}(2^{n_2})\cdot2^{n_1}\sigma_{\ell_1-1}(2^{n_1})\geq 2^{n_1}\sigma_{\ell_2-1}(2^{n_1})\cdot2^{n_2}\sigma_{\ell_1-1}(2^{n_2}). \end{align*} $$
So the result follows.
Lemma 4.8. For any
$\tau \in S_n$
, we have
$$ \begin{align*} \left|\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1})\right|<2^{\frac{n(n-1)}{2}}\cdot2^{n-1}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}. \end{align*} $$
Proof. By Lemma 4.7, we have
$$ \begin{align*} \left|\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1})\right|&\leq \prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell_i-1}(2^{\tau(i)-1})\\&\leq\prod_{i=1}^n2^{i-1}\sigma_{\ell_i-1}(2^{i-1})\\&=2^{\frac{n(n-1)}{2}}\prod_{i=1}^n\sigma_{\ell_i-1}(2^{i-1})\\&\leq2^{\frac{n(n-1)}{2}}\cdot2^{n-1}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}, \end{align*} $$
where the last inequality comes from (58).
Proposition 4.9. Let f be defined as in (59). Suppose that
$$ \begin{align} \sup_{\tau\in S_n}|f(\delta_{K,1,\tau(1)},\ldots,\delta_{K,n,\tau(n)})|<n!^{-1}\cdot \left(\frac{3}{4}\right)^{\frac{n(n-1)}{2}}\cdot2^{1-n}. \end{align} $$
Then,
$N_{K,\ell _1,\ldots ,\ell _n,\chi }$
is non-singular.
Proof. Note that
$\det N_{K,\ell _1,\ldots ,\ell _n,\chi }=$
$$ \begin{align*} &\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1})(1+\delta_{K,i,\tau(i)})\\&=\det \mathbf{N}_{K,\ell_1,\ldots,\ell_n,\chi}+\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}f(\delta_{K,1,\tau(1)},\ldots,\delta_{K,n,\tau(n)})\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1}). \end{align*} $$
We have from Lemma 4.6 that
$$ \begin{align*} \left|\det \mathbf{N}_{K,\ell_1,\ldots,\ell_n,\chi}\right|\geq\left(\frac{3}{2}\right)^{\frac{n(n-1)}{2}}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}. \end{align*} $$
Note also that
$$ \begin{align*} &\left|\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}f(\delta_{K,1,\tau(1)},\ldots,\delta_{K,n,\tau(n)})\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1})\right|\\&\leq n! \sup_{\tau\in S_n}|f(\delta_{K,1,\tau(1)},\ldots,\delta_{K,n,\tau(n)})|\left|\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell_i-1,\chi}(2^{\tau(i)-1}) \right|\\ &<\left(\frac{3}{2}\right)^{\frac{n(n-1)}{2}}\prod_{i=2}^n2^{(i-1)(\ell_i-1)}. \end{align*} $$
Therefore,
$\det N_{K,\ell _1,\ldots ,\ell _n,\chi }\neq 0$
.
Theorem 1.2 follows from the result below.
Proposition 4.10. Let
$n\geq 1$
be an integer,
$D\geq 1$
be an odd square-free integer, and
$\chi $
be a primitive Dirichlet character mod D. For
$K\gg _{n,D}1$
, if
$3\leq \ell _1<\ell _2<\cdots <\ell _n\leq \frac {K-4}{2}$
are integers such that
$\chi (-1)=(-1)^{\ell _i}$
for all
$1\leq i\leq n$
, then
$N_{K,\ell _1,\ldots ,\ell _n,\chi }$
is non-singular.
Proof. By Proposition 3.17, we have
$\delta _{K,i,\tau (i)}=o(1)$
for all
$1\leq i\leq n$
and
$\tau \in S_n$
, where
$o(1)\rightarrow 0$
when K is sufficiently large relative to n and D. Thus, Lemma 4.4 implies that
$$ \begin{align*} \sup_{\tau\in S_n}|f(\delta_{K,1,\tau(1)},\ldots,\delta_{K,n,\tau(n)})|&\leq(1+o(1))^n-1=o(1). \end{align*} $$
In particular, K can be chosen such that (64) is satisfied, showing that
$N_{K,\ell _1,\ldots ,\ell _n,\chi }$
is non-singular for
$K\gg _{n,D}1$
.
Next, we define the matrix that is used to prove Theorem 1.3. Let
$\chi _i$
be primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
and
$\chi _i(2)$
are pairwise distinct for all
$1\leq i\leq n$
. Choose the Fourier coefficients
$\mathfrak {a}_{K,\ell ,\chi _1}(1), \mathfrak {a}_{K,\ell ,\chi _2}(2),\ldots ,\mathfrak {a}_{K,\ell ,\chi _n}(2^{n-1})$
of each
$\mathfrak {F}_{K,\ell ,\chi _i}$
(
$1\leq i\leq n$
) and place them in an
$(n,n)$
-matrix
$$ \begin{align} P_{K,\ell, \chi_1,\ldots,\chi_n}:=\left[\mathfrak{a}_{K,\ell,\chi_i}(2^{j-1})\right]_{1\leq i,j\leq n}=\left[\sigma_{\ell-1,\chi_i}(2^{j-1})(1+\eta_{K,i,j})\right]_{1\leq i,j\leq n}. \end{align} $$
Define the following matrix:
$$ \begin{align} \mathbf{P}_{K,\ell,\chi_1,\ldots,\chi_n}:&=\begin{bmatrix} 1 & \sigma_{\ell-1,\chi_1}(2) & \sigma_{\ell-1,\chi_1}(2^2) & \cdots & \sigma_{\ell-1,\chi_1}(2^{n-1})\\ 1 & \sigma_{\ell-1,\chi_2}(2) & \sigma_{\ell-1,\chi_2}(2^2) & \cdots & \sigma_{\ell-1,\chi_2}(2^{n-1})\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & \sigma_{\ell-1,\chi_{n-1}}(2) & \sigma_{\ell-1,\chi_{n-1}}(2^2) & \cdots & \sigma_{\ell-1,\chi_{n-1}}(2^{n-1})\\ 1 & \sigma_{\ell-1,\chi_n}(2) & \sigma_{\ell-1,\chi_n}(2^2) & \cdots & \sigma_{\ell-1,\chi_n}(2^{n-1}) \end{bmatrix}. \end{align} $$
Lemma 4.11. Let
$\mathbf {P}_{K,\ell ,\chi _1,\ldots ,\chi _n}$
be defined as in (66). Then,
$$ \begin{align*} \left|\det \mathbf{P}_{K,\ell,\chi_1,\ldots,\chi_n}\right|\geq\left(\frac{2^{\ell+1}}{D}\right)^{\frac{n(n-1)}{2}}. \end{align*} $$
Proof. After column reductions, we have
$$ \begin{align*} \left|\det \mathbf{P}_{K,\ell,\chi_1,\ldots,\chi_n}\right|&=\left|\det\begin{bmatrix} 1 & \chi_1(2)2^{\ell-1} & \cdots & (\chi_1(2)2^{\ell-1})^{n-1}\\ \vdots & \vdots & \ddots & \vdots\\ 1 & \chi_n(2)2^{\ell-1} & \cdots & (\chi_n(2)2^{\ell-1})^{n-1} \end{bmatrix}\right|\\&=\prod_{1\leq i<j\leq n}2^{\ell-1}\left|\chi_j(2)-\chi_i(2)\right| \end{align*} $$
by the determinant of the Vandermonde matrix. Since
$\chi _i$
and
$\chi _j$
are primitive Dirichlet characters mod D such that
$\chi _i(2)\neq \chi _j(2)$
, we may assume that
$\chi _j(2)=e^{\frac {2\pi i}{\varphi (D)}r}$
and
$\chi _j(2)=e^{\frac {2\pi i}{\varphi (D)}s}$
, where
$0\leq r<s<\varphi (D)$
and
$\varphi $
is the Euler-totient function. Note that
$$ \begin{align*} |\chi_j(2)-\chi_i(2)|&=\left|e^{\frac{2\pi i}{\varphi(D)}r}-e^{\frac{2\pi i}{\varphi(D)}s}\right|\\&\geq\left|e^{\frac{2\pi i}{\varphi(D)}}-1\right|\\& \geq2\sin\left(\frac{1}{2}\cdot\frac{2\pi}{\varphi(D)}\right)\\&\geq2\cdot\frac{2}{\pi}\cdot\frac{\pi}{\varphi(D)}\\&\geq\frac{4}{D}. \end{align*} $$
It follows that
$$ \begin{align*} \left|\det \mathbf{P}_{K,\ell,\chi_1,\ldots,\chi_n}\right|&\geq\prod_{i=1}^n 2^{\ell-1}4D^{-1}=\left(\frac{2^{\ell+1}}{D}\right)^{\frac{n(n-1)}{2}}, \end{align*} $$
as desired.
Proposition 4.12. Let f be defined as in (59). Suppose that
$$ \begin{align} \sup_{\tau\in S_n}|f(\eta_{K,1,\tau(1)},\ldots,\eta_{K,n,\tau(n)})|<n!^{-1}\cdot2^{1-n}\cdot\left(\frac{4}{D}\right)^{\frac{n(n-1)}{2}}. \end{align} $$
Then,
$P_{K,\ell ,\chi _1,\ldots ,\chi _n}$
is non-singular.
Proof. Note that
$\det P_{K,\ell ,\chi _1,\ldots ,\chi _n}=$
$$ \begin{align*} &\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}\prod_{i=1}^n\sigma_{\ell-1,\chi_i}(2^{\tau(i)-1})(1+\eta_{K,i,\tau(i)})\\&\quad =\det \mathbf{P}_{K,\ell,\chi_1,\ldots,\chi_n}+\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}f(\eta_{K,1,\tau(1)},\ldots,\eta_{K,n,\tau(n)})\prod_{i=1}^n\sigma_{\ell-1,\chi_i}(2^{\tau(i)-1}). \end{align*} $$
We have from Lemma 4.11 that
$$ \begin{align*} \left|\det \mathbf{P}_{K,\ell,\chi_1,\ldots,\chi_n}\right|\geq\left(\frac{2^{\ell+1}}{D}\right)^{\frac{n(n-1)}{2}}. \end{align*} $$
Note also that for any
$\tau \in S_n$
, we have
$$ \begin{align*} \left|\prod_{i=1}^n\sigma_{\ell-1,\chi}(2^{\tau(i)-1}) \right|&\leq\prod_{i=1}^n\sigma_{\ell-1}(2^{\tau(i)-1})\\&=\prod_{i=1}^n\sigma_{\ell-1}(2^{i-1})\\& < \prod_{i=2}^n2\cdot2^{(i-1)(\ell-1)}\\&=2^{n-1}2^{\frac{(\ell-1)n(n-1)}{2}}, \end{align*} $$
which implies that
$$ \begin{align*} &\left|\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}f(\eta_{K,1,\tau(1)},\ldots,\eta_{K,n,\tau(n)})\prod_{i=1}^n\sigma_{\ell-1,\overline{\chi}}(2^{\tau(i)-1})\right|\\&\leq n! \sup_{\eta\in S_n}|f(\eta_{K,1,\tau(1)},\ldots,\eta_{K,n,\tau(n)})|2^{n-1}2^{\frac{(\ell-1)n(n-1)}{2}}\\ &<n!\cdot n!^{-1}\cdot2^{1-n}\cdot\left(\frac{4}{D}\right)^{\frac{n(n-1)}{2}}\cdot 2^{n-1}2^{\frac{(\ell-1)n(n-1)}{2}}\\&<\left(\frac{2^{\ell+1}}{D}\right)^{\frac{n(n-1)}{2}}. \end{align*} $$
Therefore,
$\det P_{K,\ell ,\chi _1,\ldots ,\chi _n}\neq 0$
.
Theorem 1.3 follows from the next result.
Proposition 4.13. Let
$D\geq 1$
be an odd square-free integer. For
$K\gg _D1$
, if
$3\leq \ell \leq \frac {K-2}{2}$
is an integer,
$\chi _1,\ldots ,\chi _n$
are primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
and
$\chi _i(2)$
are pairwise distinct for
$1\leq i\leq n$
, then
$P_{K,\ell ,\chi _1,\ldots ,\chi _n}$
is non-singular.
Proof. Since n is the number of primitive Dirichlet characters mod D that are chosen, we have
$n=O(D)$
. Again, by Proposition 3.16, we have
$\eta _{K,i,\tau (i)}=o(1)$
as for all
$1\leq i\leq n$
and
$\tau \in S_n$
, where
$o(1)\rightarrow 0$
when K is sufficiently large relative to D. Then, Lemma 4.4 implies that
$$ \begin{align*} \sup_{\tau\in S_n}|f(\eta_{K,1,\tau(1)},\ldots,\eta_{K,n,\tau(n)})|&\leq(1+o(1))^n-1=o(1). \end{align*} $$
In particular, K can be chosen such that (67) is satisfied, which shows that
$P_{K,\ell ,\chi _1,\ldots ,\chi _n}$
is non-singular as
$K\gg _{D}1$
.
Finally, we study the matrix that is used to prove Theorem 1.4. Let
$\chi _1,\ldots ,\chi _n$
be primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
and
$\chi _i(2)$
are pairwise distinct for all
$1\leq i\leq n$
. Define the coefficient matrix
$$ \begin{align} Q_{K,\ell,\chi_1,\ldots,\chi_n}:=\left[\mathfrak{b}_{K,\ell,\chi_i}(2^{j-1})\right]_{1\leq i,j\leq n}=\left[2^{j-1}\sigma_{\ell-1,\chi_i}(2^{j-1})(1+\nu_{K,i,j})\right]_{1\leq i,j\leq n}, \end{align} $$
and the matrix
$$ \begin{align} \mathbf{Q}_{K,\ell,\chi_1,\ldots,\chi_n}:&=\begin{bmatrix} 1 & 2\sigma_{\ell-1,\chi_1}(2) & 2^2\sigma_{\ell-1,\chi_1}(2^2) & \cdots & 2^{n-1}\sigma_{\ell-1,\chi_1}(2^{n-1})\\ 1 & 2\sigma_{\ell-1,\chi_2}(2) & 2^2\sigma_{\ell-1,\chi_2}(2^2) & \cdots & 2^{n-1}\sigma_{\ell-1,\chi_2}(2^{n-1})\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 2\sigma_{\ell-1,\chi_{n-1}}(2) & 2^2\sigma_{\ell-1,\chi_{n-1}}(2^2) & \cdots & 2^{n-1}\sigma_{\ell-1,\chi_{n-1}}(2^{n-1})\\ 1 & 2\sigma_{\ell-1,\chi_n}(2) & 2^2\sigma_{\ell-1,\chi_n}(2^2) & \cdots & 2^{n-1}\sigma_{\ell-1,\chi_n}(2^{n-1}) \end{bmatrix}. \end{align} $$
Lemma 4.14. Let
$\mathbf {Q}_{K,\ell ,\chi _1,\ldots ,\chi _n}$
be defined as in (69). Then,
$$ \begin{align*} \left|\det \mathbf{Q}_{K,\ell,\chi_1,\ldots,\chi_n}\right|\geq\left(\frac{2^{\ell+2}}{D}\right)^{\frac{n(n-1)}{2}}. \end{align*} $$
Proof. Since
$\det \mathbf {Q}_{K,\ell ,\chi _1,\ldots ,\chi _n}=2^{\frac {n(n-1)}{2}}\det \mathbf {P}_{K,\ell ,\chi _1,\ldots ,\chi _n}$
, the result follows from Lemma 4.11.
Proposition 4.15. Let f be defined as in (59). Suppose that
$$ \begin{align*} \sup_{\tau\in S_n}|f(\nu_{K,1,\tau(1)},\ldots,\nu_{K,n,\tau(n)})|<n!^{-1}\cdot2^{1-n}\cdot\left(\frac{4}{D}\right)^{\frac{n(n-1)}{2}}. \end{align*} $$
Then,
$Q_{K,\ell ,\chi _1,\ldots ,\chi _n}$
is non-singular.
Proof. Note that
$\det Q_{K,\ell ,\chi _1,\ldots ,\chi _n}=$
$$ \begin{align*} &\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell-1,\chi_i}(2^{\tau(i)-1})(1+\nu_{K,i,\tau(i)})\\&=\det \mathbf{Q}_{K,\ell,\chi_1,\ldots,\chi_n}+\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}f(\nu_{K,1,\tau(1)},\ldots,\nu_{K,n,\tau(n)})\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell-1,\chi_i}(2^{\tau(i)-1}). \end{align*} $$
By Lemma 4.14, we have
$$ \begin{align*} \left|\det \mathbf{Q}_{K,\ell,\chi_1,\ldots,\chi_n}\right|\geq\left(\frac{2^{\ell+2}}{D}\right)^{\frac{n(n-1)}{2}}. \end{align*} $$
For any
$\tau \in S_n$
, we have
$$ \begin{align*} \left|\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell-1,\chi}(2^{\tau(i)-1}) \right|&\leq\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell-1}(2^{\tau(i)-1})\\&=\prod_{i=1}^n2^{i-1}\sigma_{\ell-1}(2^{i-1})\\& < \prod_{i=2}^n2^{i-1}\cdot 2\cdot2^{(i-1)(\ell-1)}\\&=2^{n-1}2^{\frac{\ell n(n-1)}{2}}. \end{align*} $$
Hence,
$$ \begin{align*} &\left|\sum_{\tau\in S_n}(-1)^{\operatorname{\mathrm{sign}}(\tau)}f(\nu_{K,1,\tau(1)},\ldots,\nu_{K,n,\tau(n)})\prod_{i=1}^n2^{\tau(i)-1}\sigma_{\ell-1,\overline{\chi}}(2^{\tau(i)-1})\right|\\&\leq n! \sup_{\tau\in S_n}|f(\nu_{K,1,\tau(1)},\ldots,\nu_{K,n,\tau(n)})|2^{n-1}2^{\frac{\ell n(n-1)}{2}}\\ &<n!\cdot n!^{-1}\cdot2^{1-n}\cdot\left(\frac{4}{D}\right)^{\frac{n(n-1)}{2}}\cdot 2^{n-1}2^{\frac{\ell n(n-1)}{2}}\\&<\left(\frac{2^{\ell+2}}{D}\right)^{\frac{n(n-1)}{2}}. \end{align*} $$
Therefore,
$\det Q_{K,\ell ,\chi _1,\ldots ,\chi _n}\neq 0$
.
The next proposition can be proved the same way as the proof of Proposition 4.13.
Proposition 4.16. Let
$D\geq 1$
be an odd square-free integer. For
$K\gg _D1$
, if
$3\leq \ell \leq \frac {K-4}{2}$
is an integer,
$\chi _1,\ldots ,\chi _n$
are primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
and
$\chi _i(2)$
are pairwise distinct for
$1\leq i\leq n$
, then
$Q_{K,\ell ,\chi _1,\ldots ,\chi _n}$
is non-singular.
Thus, we have completed the proof of Theorem 1.4.
Remark 4.2. In fact, we could also consider the
$q^{p^i}$
-Fourier coefficients (p is any arbitrary fixed prime) for matrices
$P_{K,\ell ,\chi _1,\ldots ,\chi _n}$
and
$Q_{K,\ell ,\chi _1,\ldots ,\chi _n}$
, and the assumption that
$\chi _i$
’s take distinct values at
$2$
in Propositions 4.13 and 4.16 can be replaced by
$\chi _i$
’s taking distinct values at an arbitrary fixed prime p, where the results will hold for
$K\gg _{p,D}1$
.
5. Applications
5.1. Convolution sums of twisted divisor functions
Note that
$a_{K,\ell ,\chi }(n)= b_{K,\ell ,\chi }(n)=0$
for all
$n\ge 1$
when
$K<12$
or
$K=14$
because
$\dim S_K=0$
. This fact, together with Propositions 3.3 and 3.5, will give identities that evaluate convolution sums of twisted divisor functions by generalized Bernoulli numbers.
For a Dirichlet character
$\chi $
mod D, the generalized Bernoulli numbers
$B_{n,\chi }$
are defined by
$$ \begin{align*} \sum_{a=1}^D\frac{\chi(a)te^{at}}{e^{Dt}-1}=\sum_{n=0}^{\infty}B_{n,\chi}\frac{t^n}{n!}. \end{align*} $$
If
$\chi $
is primitive mod D such that
$\chi (-1)=(-1)^k$
, then the special value of
$L(s,\chi )$
is related to
$B_{k,\chi }$
by [Reference Arakawa, Ibukiyama and Kaneko1, Theorems 9.6]
$$ \begin{align} L(k,\chi)=\frac{(-1)^{k-1}(2\pi i)^k}{2k!D^k}G(\chi)B_{k,\overline{\chi}}. \end{align} $$
To obtain clean formulas, we restrict to the coefficients
$a_{K,\ell ,\chi }(1)$
and
$b_{K,\ell ,\chi }(1)$
for
$D=p$
. Readers may compare the following results with the classical ones (see, e.g., [Reference Royer24]).
Theorem 5.1. Let
$p>2$
be a prime number and
$\chi $
be a primitive Dirichlet character mod p.
-
1. If
$\chi (-1)=-1$
, then we have -
2. If
$\chi (-1)=1$
, then we have
Proof. Note that (5) and (70) imply that
$$ \begin{align} \sigma_{k-1,\chi}(0)=-\frac{B_{k,\chi}}{2k}. \end{align} $$
Using Proposition 3.3 for
$D=p$
, and (71), we get
where
$k=K-\ell $
. Note that if
$$ \begin{align*}(\ell,k)\in \mathcal{S}:=\{(3,3),(3,5),(3,7),(3,11);(4,4),(4,6),(4,10);(5,5),(5,9);(7,7) \},\end{align*} $$
then
$0\equiv \mathcal {F}_{K,\ell ,\chi }\in S_{K}$
since for those values of
$(\ell ,k),$
we have
$\dim S_K=0.$
It follows that for all
$(\ell ,k)\in \mathcal {S,}$
we have
Writing out (72) explicitly gives the result.
Theorem 5.2. Let
$p>2$
be a prime number, and let
$\chi $
be a primitive Dirichlet character mod p.
-
1. If
$\chi (-1)=-1$
, then -
2. If
$\chi (-1)=1$
, then (73)
Proof. Applying Proposition 3.5 for
$D=p$
, we get
where
$k=K-2-\ell $
. Note that if
$(\ell ,k)\in \mathcal {T}:=\{(3,5),(3,9),(4,8),(5,7)\},$
then
$0\equiv \mathcal {G}_{K,\ell ,\chi }\in S_K$
since for those values of
$(\ell ,k),$
we have
$\dim S_K=0$
. It follows that for all
$(\ell ,k)\in \mathcal {T,}$
we have
which gives the result.
Remark 5.1. Some small cases (
$D=1,3,5,7,11,13,15$
) of Theorems 5.1 and 5.2 have been numerically verified by [Reference Ross23] (see also Section 6).
5.2. Non-vanishing of twisted central Lvalues
We now focus on the case when
$\chi $
is real, or equivalently is quadratic. As
$G(\chi )^2=\chi (-1) D$
in this case, for a normalized Hecke eigenform
$f\in S_{K}$
, the functional equation (3) becomes
$$ \begin{align*} \Lambda(f,\chi,s)=(-1)^{K/2}\chi(-1)\Lambda(f,\chi,K-s). \end{align*} $$
Note that if
$(-1)^{K/2}\chi (-1)<0,$
then
$L(f,\chi ,K/2)$
is trivially zero. Some evidence has been provided for the non-vanishing of
$L(f,\chi ,K/2)$
when
$(-1)^{K/2}\chi (-1)>0$
. For example, Lau and Tsang [Reference Lau and Tsang15] showed that for
$1\leq D\leq K^{1/6}/(\log K)^5$
, we have
$$ \begin{align*} \#\{f\in\mathcal{H}_K:L(f,\chi,K/2)\neq0\}\gg|1+\epsilon_{K}(\chi)|^2\frac{D}{\varphi(D)}\frac{K}{(\log K)^2}, \end{align*} $$
where
$\epsilon _K(\chi )=i^KG(\chi )^2/D$
, and
$\mathcal {H}_K$
denotes the set of normalized Hecke eigenforms in
$S_{K}$
.
As an application of our methods, we show that Maeda’s conjecture implies that
$L(f,\chi ,K/2)$
is non-vanishing as
$K\geq 10D+2$
. Following [Reference Maeda18], we first review Maeda’s conjecture. Let
$f(z)=\sum _{n=1}^{\infty }a_f(n)q^n\in S_K$
be a normalized Hecke eigenform. Denote by
$K_f$
the field generated over
$\mathbb {Q}$
by the coefficients
$a_f(n)$
for all n. Let
$G(f)$
be the Galois group of the Galois closure of
$K_f$
over
$\mathbb {Q}$
. It is well-known that
$K_f$
is a number field, and for every
$\sigma \in G(f)$
,
$$ \begin{align*} f^{\sigma}(z):=\sum_{n=1}^{\infty}a_{f}(n)^{\sigma}q^n \end{align*} $$
is also a normalized Hecke eigenform in
$S_K$
. Maeda’s conjecture can be formulated as follows.
Conjecture 5.3 [Reference Maeda18, p. 306]
Let
$f\in S_K$
be a normalized Hecke eigenform. Then,
-
(Ha)
$\operatorname {\mathrm {Span}}\{f^{\sigma }:\sigma \in G(f)\}=S_K$
; -
(Hb)
$G_f$
is isomorphic to the symmetric group of degree
$\dim S_K$
.
Proposition 5.4. Let
$\mathcal {H}_K$
denote the set of normalized Hecke eigenforms in
$S_K$
. Assume
$(H_a)$
holds true for
$S_K$
. Suppose there exists some
$f\in \mathcal {H}_K$
such that
$L(f,\chi ,K/2)\neq 0$
. Then,
$L(g,\chi ,K/2)\neq 0$
for all
$g\in \mathcal {H}_K$
.
Proof. Since
$L(f,\chi ,K/2)\neq 0$
, we have
$L(f^{\sigma },\chi ^{\sigma },K/2)=L(f^{\sigma },\chi ,K/2)\neq 0$
for all
$\sigma \in G(f)$
by Shimura’s result [Reference Shimura25, Theorem 1]. For any
$g\in \mathcal {H}_K$
, there exists some
$\tau \in G(f)$
such that
$g=f^{\tau }$
by assumption. Hence,
$L(g,\chi ,K/2)=L(f^{\tau },\chi ,K/2)\neq 0$
.
The following result can be viewed as a generalization of [Reference Conrey and Farmer4, Theorem 1].
Theorem 5.5. Let
$D\geq 1$
be an odd square-free integer and
$\chi $
be a primitive quadratic Dirichlet character D such that
$(-1)^{K/2}\chi (-1)>0$
. If
$(H_a)$
holds for
$S_K$
and
$K\geq 10D+2$
, then for every Hecke eigenform
$f\in S_K$
,
$$ \begin{align*} L(f,\chi,K/2)\neq 0. \end{align*} $$
Proof. As mentioned in Remark 2.1, (19) extends to
$k=\ell =\frac {K}{2}$
:
$$ \begin{align*} \langle f,\mathcal{F}_{K,k,\chi}\rangle=\frac{\Gamma(K-1)(k-1)!D^{k}}{(4\pi)^{K-1}(2\pi i)^{k}\overline{G(\chi)}}L(f,K-1)L(f,\chi,K/2). \end{align*} $$
Since
$L(f,K-1)\neq 0$
, we have
$$ \begin{align*} L(f,\chi,K/2)=0\quad\mathrm{if~and~only~if}\quad\langle f,\mathcal{F}_{K,k,\chi}\rangle=0. \end{align*} $$
By Proposition 5.4, it suffices to show that
$\mathcal {F}_{K,k,\chi }\not \equiv 0$
for
$K\geq 10D+2$
, which is proved in Proposition 5.6.
To show that
$\mathcal {F}_{K,k,\chi }\not \equiv 0$
, it suffices to prove that at least one Fourier coefficient of
$\mathcal {F}_{K,k,\chi }$
is non-vanishing. It is convenient to consider the q-Fourier coefficient
$a_{K,k,\chi }(1)$
of
$\mathcal {F}_{K,k,\chi }$
.
Proposition 5.6. Let
$D\geq 1$
be an odd square-free integer and
$\chi $
be a primitive quadratic Dirichlet character D such that
$(-1)^{K/2}\chi (-1)>0$
. If
$K\geq 10D+2$
, then
$a_{K,k,\chi }(1)\neq 0$
.
Proof. By Proposition 3.3, Corollary 3.1 (and Remark 3.1), and the fact that
$\chi =\overline {\chi }$
, we have
$$ \begin{align*} a_{K,k,\chi}(1) =2\sigma_{k-1,\chi}(0)+\sum_{\substack{D=D_1D_2\\D_2\neq 1 }}\chi_2(-1)^{-1}\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=D_2}}\sigma_{k-1,\chi_1,\chi_2}(a_1)\sigma_{k-1,\chi_1,\chi_2}(a_2)-\frac{2\sigma_{k-1,\chi}(0)^2}{\zeta(1-K)}. \end{align*} $$
Thus, we can write
$a_{K,k,\chi }(1)=2\sigma _{k-1,\chi }(0)(1+\varepsilon _{K,D})$
, where
$$ \begin{align*} \varepsilon_{K,D}=\frac{\sigma_{k-1,\chi}(0)^{-1}}{2}\sum_{\substack{D=D_1D_2\\D_2\neq 1 }}\chi_2(-1)^{-1}\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=D_2}}\sigma_{k-1,\chi_1,\chi_2}(a_1)\sigma_{\ell-1,\chi_1,\chi_2}(a_2)-\frac{\sigma_{k-1,\chi}(0)}{\zeta(1-K)}. \end{align*} $$
To show
$a_{K,k,\chi }(1)\neq 0$
, it suffices to show that
$|\varepsilon _{K,D}|<1$
. Note that (39) implies that
$$ \begin{align} \left|\frac{\sigma_{k-1,\chi}(0)}{\zeta(1-K)}\right|&\leq \frac{e\zeta(k)}{2\sqrt{2\pi}}\left(\frac{(k-1)D}{K-1}\right)^{k-1/2}\left(\frac{2\pi e}{K-1}\right)^{k} \nonumber\\&\leq\frac{e\zeta(k)}{2\sqrt{2\pi}}\sqrt{\frac{K-1}{(k-1)D}}\left(\frac{(K-2)\pi eD}{(K-1)^2}\right)^k \nonumber\\&\leq\frac{e\zeta(k)}{2\sqrt{2\pi}}\sqrt{\frac{3}{D}}\left(\frac{\pi e D}{K-1}\right)^k. \end{align} $$
Using (44) for
$\ell =k$
and
$n=1$
, we get
$$ \begin{align} \left|\sum_{\substack{D=D_1D_2\\D_2\neq 1 }}\overline{\chi}_2(-1)\sum_{\substack{a_1,a_2\geq0\\a_1+a_2=D_2}}\sigma_{k-1,\chi_1,\chi_2}(a_1)\sigma_{k-1,\chi_1,\chi_2}(a_2)\right|\leq2\zeta(k-1)^2\zeta(K-1)\left(\frac{D}{2}\right)^{K-1}. \end{align} $$
Note also that (41) implies that
$$ \begin{align} \left|\frac{\sigma_{k-1,\overline{\chi}}(0)^{-1}}{2}\right|\leq\frac{1}{2\sqrt{e}(2-\zeta(k))}\left(\frac{2\pi e}{(k-1)D}\right)^{k-1/2}. \end{align} $$
$$ \begin{align*} |\varepsilon_{K,D}|\leq &\frac{\zeta(k-1)^2\zeta(K-1)}{\sqrt{e}(2-\zeta(k))}\left(\frac{\pi eD}{K-2}\right)^{\frac{K-1}{2}}+ \frac{e\zeta(k)}{2\sqrt{2\pi}}\sqrt{\frac{3}{D}}\left(\frac{\pi e D}{K-1}\right)^k. \end{align*} $$
If
$K-2\geq 10D$
, then
$$ \begin{align*} |\varepsilon_{K,D}|<\frac{\zeta(5)^2\zeta(11)}{\sqrt{e}(2-\zeta(6))}\left(\frac{\pi e}{10}\right)^{\frac{11}{2}}+ \frac{e\zeta(6)}{2\sqrt{2\pi}}\sqrt{3}\left(\frac{\pi e }{10}\right)^6\approx0.65<1, \end{align*} $$
which completes the proof.
6. Conjectures
We believe that twisted periods with fixed character but different indices span the whole vector space
$S_K^{\ast }$
, which follows from the conjectures below.
Conjecture 6.1. Let
$3\leq \ell _1<\ell _2<\cdots <\ell _n\leq \frac {K-2}{2}$
be integers,
$D\geq 1$
be an integer, and
$\chi $
be a primitive Dirichlet character such that
$\chi (-1)=(-1)^{\ell _i}$
for all
$1\leq i\leq n$
. If
$n\leq \dim S_K$
, then
$\{r_{\ell _i-1,\chi }\}_{i=1}^{n}$
is linearly independent.
Conjecture 6.2. Let
$3\leq \ell _1<\ell _2<\cdots <\ell _n\leq \frac {K-4}{2}$
be integers, let
$D\geq 1$
be an integer, and let
$\chi $
be a primitive Dirichlet character such that
$\chi (-1)=(-1)^{\ell _i}$
for all
$1\leq i\leq n$
. If
$n\leq \dim S_K$
, then
$\{r_{\ell _i,\chi }\}_{i=1}^{n}$
is linearly independent.
The above conjectures are analogs of [Reference Lei, Ni and Xue17, Conjecture 1.5] and [Reference Lei, Ni and Xue16, Conjecture 4.5], respectively. We verified Conjectures 6.1 and 6.2 for
$K,D\leq 40$
[Reference Ross23] by checking the non-singularity of the matrices
$$ \begin{align*} \mathcal{C}_1:=& \begin{bmatrix} a_{K,\ell_1,\chi}(1) & a_{K,\ell_1,\chi}(2) & \cdots & a_{K,\ell_1,\chi}(n) \\ a_{K,\ell_2,\chi}(1) & a_{K,\ell_2,\chi}(2) & \cdots & a_{K,\ell_n,\chi}(n) \\ \vdots &\vdots &\ddots &\vdots\\a_{K,\ell_n,\chi}(1) & a_{K,\ell_n,\chi}(2) & \cdots & a_{K,\ell_n,\chi}(n) \end{bmatrix}\end{align*} $$
and
$$ \begin{align*} \mathcal{C}_2:=&\begin{bmatrix} b_{K,\ell_1,\chi}(1) & b_{K,\ell_1,\chi}(2) & \cdots & b_{K,\ell_1,\chi}(n) \\ b_{K,\ell_2,\chi}(1) & b_{K,\ell_2,\chi}(2) & \cdots & b_{K,\ell_n,\chi}(n) \\ \vdots &\vdots &\ddots &\vdots\\b_{K,\ell_n,\chi}(1) & b_{K,\ell_n,\chi}(2) & \cdots & b_{K,\ell_n,\chi}(n) \end{bmatrix}, \end{align*} $$
where
$a_{K,\ell ,\chi }(n)$
and
$b_{K,\ell ,\chi }(n)$
denote the
$q^n$
-Fourier coefficient of
$\mathcal {F}_{K,\ell ,\chi }(z)$
and
$\mathcal {G}_{K,\ell ,\chi }(z)$
, respectively.
Let
$N(D)$
denote the number of primitive Dirichlet characters mod D. We also believe that twisted periods with fixed index but different twists mod D span
$S_K^{\ast }$
as long as
$N(D)\geq \dim S_K$
, which are corollaries of the following conjectures.
Conjecture 6.3. Let
$D\geq 1$
and
$3\leq \ell \leq \frac {K-2}{2}$
be integers, and
$\chi _1,\ldots ,\chi _n$
be distinct primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
. If
$n\leq \dim S_K$
, then the set of twisted periods
$\{r_{\ell -1,\chi _i}\}_{i=1}^n$
on
$S_K$
is linearly independent.
Conjecture 6.4. Let
$D\geq 1$
and
$3\leq \ell \leq \frac {K-4}{2}$
be integers, and
$\chi _1,\ldots ,\chi _n$
be distinct primitive Dirichlet characters mod D such that
$\chi _i(-1)=(-1)^{\ell }$
. If
$n\leq \dim S_K$
, then the set of twisted periods
$\{r_{\ell ,\chi _i}\}_{i=1}^n$
on
$S_K$
is linearly independent.
We also verified Conjectures 6.3 and 6.4 for
$K,D\leq 40$
[Reference Ross23] by checking matrices
$$ \begin{align*} \mathcal{C}_3:=\begin{bmatrix} a_{K,\ell,\chi_1}(1) & a_{K,\ell,\chi_1}(2) & \cdots & a_{K,\ell,\chi_1}(n) \\ a_{K,\ell,\chi_2}(1) & a_{K,\ell,\chi_2}(2) & \cdots & a_{K,\ell,\chi_2}(n) \\ \vdots &\vdots &\ddots &\vdots\\a_{K,\ell,\chi_n}(1) & a_{K,\ell,\chi_n}(2) & \cdots & a_{K,\ell,\chi_n}(n) \end{bmatrix}\end{align*} $$
and
$$ \begin{align*} \mathcal{C}_4:=\begin{bmatrix} b_{K,\ell,\chi_1}(1) & b_{K,\ell,\chi_1}(2) & \cdots & b_{K,\ell,\chi_1}(n) \\ b_{K,\ell,\chi_2}(1) & b_{K,\ell,\chi_2}(2) & \cdots & b_{K,\ell,\chi_2}(n) \\ \vdots &\vdots &\ddots &\vdots\\b_{K,\ell,\chi_n}(1) & b_{K,\ell,\chi_n}(2) & \cdots & b_{K,\ell,\chi_n}(n) \end{bmatrix} \end{align*} $$
are non-singular.
Acknowledgements
The authors thank the anonymous referee for the detailed comments and insightful advice that have improved the exposition of this article. The authors also thank Erick Ross for numerically verifying Conjectures 6.1–6.4.
Funding
Research of H.X. is supported by the Simons Foundation Grant MPS-TSM-00007911.
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