2.1 Zagier’s Proof

We begin by sketching his proof, which relies on the following theorem.

Theorem (Zagier [Reference Zagier33]).

We have that

is a harmonic Maass form of weight $3/2$ on $\Gamma _0(4)$ , where and $\Gamma (s; x)$ is the incomplete Gamma function. Its holomorphic part is the Fourier series

Zagier uses a sequence of modular forms he constructs from $\mathcal {H}(\tau )$ and Jacobi’s weight 1/2 theta function

(2.1)

To define these modular forms, he requires Atkin’s U-operator defined by

(2.2)

and the Rankin–Cohen bracket differential operators. For modular forms f and g (possibly non-holomorphic), with weights k and l, respectively, these operators are defined by

(2.3)

where . These functions are weight $2\nu +k+l$ (possibly non-holomorphic) modular forms, which one can project to obtain a holomorphic modular form via an integral map $\pi _{\mathrm {hol}}.$

Zagier studies the resulting sequence of modular forms $\pi _{\mathrm {hol}}([\mathcal {H}, \theta ]_\nu |U_4)$ , where $\nu \geq 1$ . He computes them in two ways. The first method is combinatorial, and it uses the identity (for example, see [Reference Mertens20, Reference Mertens21])

$$\begin{align*}\pi_{\mathrm{hol}}([\mathcal{H}, \theta]_\nu|U_4) = [\mathcal{H}^+, \theta]_\nu|U_4 + 2 \binom{2\nu}{\nu} \sum_{n=1}^\infty \lambda_{2\nu+1}(n) q^n. \end{align*}$$

A straightforward brute force calculation with (2.3) gives

(2.4) $$ \begin{align} [\mathcal{H}^+, \theta]_\nu |U_4 = \binom{2\nu}{\nu} \sum_{n=0}^\infty \left(\sum_{r \in \mathbb{Z}} p_{2\nu+2}(r,n) H(4n-r^2) \right) q^n. \end{align} $$

Therefore, the nth coefficient of $\pi _{\mathrm {hol}}([\mathcal {H}, \theta ]_\nu |U_4)$ is

(2.5) $$ \begin{align} \binom{2\nu}{\nu} \left(\sum_{r \in \mathbb{Z}} p_{2\nu+2}(r,n) H(4n-r^2) + 2\lambda_{2\nu+1}(n) \right). \end{align} $$

As an alternate calculation, Zagier combines (for example, see [Reference Eichler and Zagier13, Theorem 5.5]) the Rankin–Cohen bracket operators with Hecke–Petersson theory. As each $\pi _{\mathrm {hol}}([\mathcal {H}, \theta ]_\nu |U_4)$ is a cusp form, we have

$$ \begin{align*} \pi_{\mathrm{hol}}([\mathcal{H}, \theta]_\nu |U_4) = \sum_{j=1}^{d_{2\nu+2}} a_j f_j, \end{align*} $$

where the $f_j$ ’s form a basis of Hecke eigenforms for $S_{2\nu +2}$ . In particular, we have $T_n f_j = c_{f_j}(n) f_j,$ where

$$ \begin{align*}f_j(\tau)=q+\sum_{n\geq 2} c_{f_j}(n)q^n.\end{align*} $$

To compute the $a_j,$ he expresses $\mathcal {H}(\tau )$ in terms of Eisenstein series (see [Reference Ono and Saad26, Section 2.2] or [Reference Hirzebruch and Zagier14, Ch. 2]), which allows him to use the method of unfolding and the Rankin–Selberg method to derive the Petersson inner product identity (for example, see [Reference Bringmann, Folsom, Ono and Rolen4, Ch. 6.3])

$$ \begin{align*} a_j \langle f_j, f_j \rangle = \langle \pi_{\mathrm{hol}}([\mathcal{H}, \theta]_\nu |U_4), f_j \rangle = -2 \binom{2\nu}{\nu} \langle f_j, f_j \rangle. \end{align*} $$

For each j, this gives $a_j=-2\binom {2\nu }{\nu }.$ Therefore, the nth coefficient of $\pi _{\mathrm {hol}}([\mathcal {H}, \theta ]_\nu |U_4)$ is $-2 \binom {2\nu }{\nu }\cdot \mathrm {Tr}(n; 2\nu +2),$ which when equated with (2.5) gives the Eichler–Selberg trace formula.

2.2 Proofs of Theorems 1.1–1.3

Zagier’s proof begins with the fact that $\mathcal {H}^+(\tau )$ is the holomorphic part of a weight 3/2 harmonic Maass form. In 2002, Zagier [Reference Zagier34] greatly generalized this fact.

Theorem 5 of [Reference Zagier34].

For positive integers m, we have that

(2.6)

is a weakly holomorphic modular form of weight $3/2$ on $\Gamma _0(4)$ .

Proof of Theorem 1.1.

Emulating Zagier’s proof of the Eichler–Selberg trace formula, we replace $\mathcal {H}^+(\tau )$ in (2.4) with the $g_m(\tau )$ . Namely, we define

By the combinatorial calculation that gave (2.4), we obtain the earlier definition (1.5)

$$ \begin{align*} \mathcal{G}_{m,\nu}(\tau) = -\frac{1}{2} \sum_{n \gg -\infty} \sum_{r \in \mathbb{Z}} p_{2\nu+2}(r,n) \mathbf{t}_m(4n-r^2) q^n. \end{align*} $$

Furthermore, the theory of Rankin–Cohen brackets in this setting (see [Reference Eichler and Zagier13, Theorem 5.5]) implies that $\mathcal {G}_{m,\nu }(\tau )$ is a weakly holomorphic modular form in $M_{2\nu +2}^!.$

Proof of Theorem 1.2.

The space of weight 2 holomorphic modular forms is $M_2 = \{0\}$ and

$$ \begin{align*}Dj_{-n}(\tau) = n q^n + O(q) \in M_2^!. \end{align*} $$

Therefore, we have

$$ \begin{align*} \mathcal{G}_{m,0}(\tau) +\frac{1}{2} \sum_{-\frac{m^2}{4} \le n < 0} \frac{1}{n} \left(\sum_{r \in \mathbb{Z}} \mathbf{t}_m(4n-r^2)\right) Dj_{-n}(\tau) = 0. \end{align*} $$

The first claim follows from (1.6). By comparing the nth coefficients, the second claim is obtained.

Proof of Theorem 1.3.

For $\nu> 0$ , we note that

(2.7) $$ \begin{align} \mathcal{G}_{m,\nu}(\tau) + \frac{1}{2} \sum_{-\frac{m^2}{4} \le n \le 0} \sum_{r \in \mathbb{Z}} p_{2\nu+2}(r,n) \mathbf{t}_m(4n-r^2) P_{2\nu+2, n}(\tau) \end{align} $$

is a cusp form. We are merely cancelling the poles at infinity with Poincaré series that satisfy (1.8), and we capture the constant term with Eisenstein series $P_{2\nu +2,0}(\tau )=E_{2\nu +2}(\tau )=1+\cdots .$ For $\nu \in \{1, 2, 3, 4, 6\}$ , the space of cusp forms $S_{2\nu +2}=\{0\}$ is trivial. Therefore, the theorem follows from the identity

(2.8) $$ \begin{align} p_{2\nu+2} \left(r, \frac{r^2-\kappa^2}{4}\right) = \frac{(\kappa - r)^{2\nu+1} + (\kappa + r)^{2\nu+1}}{2^{2\nu+1} \kappa}. \end{align} $$

3.1 The Weil representation

Let $\mathcal {O}(\mathbb {H})$ be the set of all holomorphic functions $\phi : \mathbb {H} \to \mathbb {C}$ . For $z \in \mathbb {C} \setminus \{0\}$ , we take the principal branch of $z^{1/2}$ as $\arg (z^{1/2}) \in (-\pi /2, \pi /2]$ . For an integer $k \in \mathbb {Z}$ , we put $z^{k/2} = (z^{1/2})^k$ . For $n \in \mathbb {Z}_{\ge 0}$ , we put , and .

The metaplectic group $\operatorname {\mathrm {Mp}}_2(\mathbb {R})$ is a group defined by

where the group operation is

As usual, we have , and for any $\gamma = \bigl (\begin {smallmatrix}a & b \\ c & d\end {smallmatrix}\bigr ) \in \operatorname {\mathrm {SL}}_2(\mathbb {R})$ , we define $j(\gamma , \tau ) = c\tau +d$ and $\widetilde {\gamma } = \left (\bigl (\begin {smallmatrix}a & b \\ c & d\end {smallmatrix}\bigr ), j(\gamma , \tau )^{1/2} \right ) \in \operatorname {\mathrm {Mp}}_2(\mathbb {R})$ . Let $\operatorname {\mathrm {Mp}}_2(\mathbb {Z})$ be the inverse image of $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ under the projection $\operatorname {\mathrm {Mp}}_2(\mathbb {R}) \to \operatorname {\mathrm {SL}}_2(\mathbb {R})$ . As usual, we let and It is well known that $\operatorname {\mathrm {Mp}}_2(\mathbb {Z})$ is generated by $\widetilde {T}$ and $\widetilde {S}$ , (see [Reference Bruinier6, p.16]) and its center is generated by

$$\begin{align*}\widetilde{-I} = \widetilde{S}^2 = (\widetilde{S} \widetilde{T})^3 = \left(\begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix}, i \right). \end{align*}$$

Moreover, we let , representing the metaplectic stabilizer for the cusp at infinity.

We recall the Weil representation,Footnote ¹ the unitary representation $\rho : \operatorname {\mathrm {Mp}}_2(\mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {C})$ defined by

(3.1)

We note that $\rho (\widetilde {-I}) = \rho (\widetilde {S}^2) = -i I$ . We let $\rho ^* : \operatorname {\mathrm {Mp}}_2(\mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {C})$ be the dual representation of $\rho $

We recall an explicit formula for $\rho (\widetilde {\gamma })$ , which is easily derived from work of both Shintani [Reference Shintani29, Proposition 1.6] and Bruinier [Reference Bruinier6, Proposition 1.1], where for odd integers d, we let

(3.2)

Proposition 3.1. For $c \ge 0$ , we have

$$ \begin{align*} \rho(\widetilde{\gamma}) = \begin{cases} \displaystyle{\frac{\epsilon_c}{1+i} \left(\frac{a}{c}\right) \begin{pmatrix}1 & i^{cd} \\ i^{ac} & -i^{(a+d)c}\end{pmatrix}} &\text{if } c \equiv 1\quad \pmod{2},\\ \displaystyle{\epsilon_a^{-1} \left(\frac{c}{a}\right) \begin{pmatrix}0 & i^{ab} \\ 1 & 0\end{pmatrix}} &\text{if } c \equiv 2\quad \pmod{4},\\ \displaystyle{\epsilon_a^{-1} \left(\frac{c}{a}\right) \begin{pmatrix}1 & 0 \\ 0 & i^{ab}\end{pmatrix}} &\text{if } c \equiv 0\quad \pmod{4}. \end{cases} \end{align*} $$

We now give the definition of a vector-valued modular form that transforms under the Weil representation. If $k \in \frac {1}{2} \mathbb {Z}$ and $f: \mathbb {H} \to \mathbb {C}^2$ . For $(\gamma , \phi (\tau )) \in \operatorname {\mathrm {Mp}}_2(\mathbb {Z}),$ then we define the slash operator

We say that $f: \mathbb {H} \to \mathbb {C}^2$ is a weight k (vector-valued) modular form with respect to $\rho $ if

$$\begin{align*}f|_{k,\rho} (\gamma, \phi) = f \end{align*}$$

for every $(\gamma , \phi ) \in \operatorname {\mathrm {Mp}}_2(\mathbb {Z})$ . We define them for $\rho ^*$ in a similar manner.

3.2 Jacobi’s theta functions

For later use, we recall the Jacobi theta functions (for example, see [Reference Eichler and Zagier13, Section 5]) in this context. If we set , where we have

(3.3)

and The specialization $\Theta (\tau , 0)$ is a weight 1/2 vector-valued modular form with respect to $\rho $ , and in general is a (vector-valued) Jacobi form, which for $(\gamma , \phi )\in \operatorname {\mathrm {Mp}}_2(\mathbb {Z}),$ in this case, means that

(3.4)

4.1 The Whittaker functions

Let $M_{\mu , \nu }(z)$ and $W_{\mu , \nu }(z)$ be the Whittaker functions (for example, see [Reference Whittaker and Watson30, Ch. 16] and [Reference Magnus, Oberhettinger and Soni17, 24]). The next two lemmas are crucial for constructing Maass–Poincaré series.

Lemma 4.1 [Reference Magnus, Oberhettinger and Soni17, 7.2.1], [24, 13.15.19].

For positive integers $n,$ we have

$$\begin{align*}\frac{\mathrm{d}^n}{\mathrm{d} z^n} \left(e^{-z/2} z^{-\nu-1/2} M_{\mu, \nu}(z) \right) = (-1)^n \frac{(\mu + \nu + 1/2)^{\overline{n}}}{(2\nu+1)^{\overline{n}}} e^{-z/2} z^{-\nu - n/2-1/2} M_{\mu+n/2, \nu+n/2}(z). \end{align*}$$

Lemma 4.2 [Reference Magnus, Oberhettinger and Soni17, 7.5.1], [24, 13.23.1].

For $\operatorname {\mathrm {Re}}(\nu + \alpha + 1/2)> 0$ and $2\operatorname {\mathrm {Re}}(z)> \beta > 0$ , we have

$$ \begin{align*} \int_0^\infty e^{-zt} t^{\alpha-1} M_{\mu, \nu}(\beta t) \mathrm{d} t &= \frac{\beta^{\nu+1/2} \Gamma\left(\nu+\alpha+\frac{1}{2}\right)}{\left(z+\frac{\beta}{2}\right)^{\nu+\alpha+1/2}}\cdot {}_2F_1 \left(\nu-\mu+\frac{1}{2}, \nu+\alpha+\frac{1}{2}; 2\nu+1; \frac{2\beta}{\beta+2z}\right), \end{align*} $$

where ${}_2F_1(a,b; c; z)$ is the Gaussian hypergeometric function.

For $n \in \mathbb {Z}$ , $k \in \frac {1}{2} \mathbb {Z}$ , $y> 0$ , and $s \in \mathbb {C}$ , we define the modified Whittaker functions

(4.1)

(4.2)

The special values of these functions at $s = k/2$ play a crucial role in the construction of the Maass–Poincaré series. To this end, for $n < 0$ , we have

(4.3) $$ \begin{align} \mathcal{M}_{k,n} \left(y, \frac{k}{2}\right) &= \Gamma(k)^{-1} e^{-2\pi ny}. \end{align} $$

As for the $\mathcal {W}$ -function, we have

(4.4) $$ \begin{align} \mathcal{W}_{k,n} \left(y, \frac{k}{2}\right) = \begin{cases} \Gamma(k)^{-1} n^{k-1} e^{-2\pi ny} &\text{if } n> 0,\\ 0 &\text{if } n \le 0 \end{cases} \end{align} $$

(see [Reference Magnus, Oberhettinger and Soni17, 7.2.4]). Moreover, we note that, ([Reference Magnus, Oberhettinger and Soni17, 7.6.1], [24, 13.14]),

(4.5) $$ \begin{align} \begin{split} W_{\mu, \nu}(y) &\sim e^{-y/2} y^\mu \quad (y \to \infty),\\ M_{\mu, \nu}(y) &= y^{\nu+1/2} (1+ O(y)) \quad (y \to 0). \end{split} \end{align} $$

4.2 Kloosterman sums

The Fourier expansions of the Maass–Poincaré series require Kloosterman sums, which we recall here. For $k \in \frac {1}{2}\mathbb {Z} \setminus \mathbb {Z}$ , $m, n \in \mathbb {Z}$ , and $c>0$ with $c \equiv 0\ \pmod {4}$ , we define the half-integral weight Kloosterman sum by

(4.6)

where $\overline {d} \in \mathbb {Z}/c\mathbb {Z}$ satisfies that $d \overline {d} \equiv 1\ \pmod {c}$ . The condition $d\ (c)^*$ means that d runs over $d \in \mathbb {Z}/c\mathbb {Z}$ such that $(c,d) = 1$ . We note that the Kloosterman sums satisfy

(4.7) $$ \begin{align} K_{k+2}(m,n,c) = K_k(m,n,c) \quad \text{ and } \quad K_{3/2}(m,n,c) = -i K_{1/2}(-m,-n,c). \end{align} $$

We now relate the Weil representation to such Kloosterman sums. For notational convenience, we let

$$\begin{align*}\rho(\widetilde{\gamma}) = \begin{pmatrix}\rho(\widetilde{\gamma})_{00} & \rho(\widetilde{\gamma})_{01} \\ \rho(\widetilde{\gamma})_{10} & \rho(\widetilde{\gamma})_{11}\end{pmatrix}. \end{align*}$$

Then the following sum formula holds for each entry of $\rho (\widetilde {\gamma })$ .

Proposition 4.3. If $\alpha , \beta \in \{0,1\}$ and m and n satisfy $m \equiv -\alpha\ \pmod {4}$ and $n \equiv -\beta\ \pmod {4},$ then for every positive integer c, we have

$$ \begin{align*} \frac{1}{4} \left(1 + \left(\frac{4}{c}\right) \right) K_{3/2}(m,n,4c)= \sum_{d\ (c)^*} \rho(\widetilde{\gamma})_{\alpha \beta} \mathbf{e}\left(\frac{ma + nd}{4c}\right), \end{align*} $$

where we take any $\gamma = \bigl (\begin {smallmatrix}a & b \\ c & d\end {smallmatrix}\bigr ) \in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ for which $(c,d)$ forms its bottom row.

Proof. First, we check that the right-hand side is well defined. Let $R_{\alpha \beta }(\gamma )$ denote its summand. It suffices to show that $R_{\alpha \beta }(T^j \gamma T^l) = R_{\alpha \beta }(\gamma )$ holds for any $j, l \in \mathbb {Z}$ . Since $\widetilde {T^j \gamma T^l} = \widetilde {T}^j \widetilde {\gamma } \widetilde {T}^l$ holds, we have

$$\begin{align*}R_{\alpha \beta}(T^j \gamma T^l) = i^{\alpha j + \beta l} \rho(\widetilde{\gamma})_{\alpha \beta} \mathbf{e} \left(\frac{ma+nd}{4c}\right) \mathbf{e} \left(\frac{mj+nl}{4}\right) = R_{\alpha \beta}(\gamma). \end{align*}$$

Next, for each $\gamma = \bigl (\begin {smallmatrix}a & b \\ c & d\end {smallmatrix}\bigr ) \in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ with $c> 0$ , we prove the refined equation

(4.8) $$ \begin{align} \rho(\widetilde{\gamma})_{\alpha \beta} = \frac{1}{4} \left(1 + \left(\frac{4}{c}\right) \right) \sum_{\substack{\delta\ (4c) \\ \delta \equiv d\ (c) \\ \delta \equiv 1\ (2)}} \left(\frac{c}{\delta}\right) \epsilon_{\delta}^{-1} \mathbf{e} \left(\frac{a - \overline{\delta}}{4c}\right)^\alpha \mathbf{e} \left(\frac{d-\delta}{4c}\right)^\beta, \end{align} $$

where $\overline {\delta }$ is the inverse of $\delta $ in $(\mathbb {Z}/4c\mathbb {Z})^\times $ . This immediately implies the proposition.

To confirm (4.8), let and ( $j = 0,1,2,3$ ). For simplicity, let $\rho "(\gamma )_{\alpha \beta }$ denote the right-hand side of (4.8) and show that $\rho "(\gamma )_{\alpha \beta } = \rho (\widetilde {\gamma })_{\alpha \beta }$ . If $\delta $ is odd, then we can easily check that

$$\begin{align*}\overline{\delta} = \begin{cases} a(1+b_j c) &\text{if } c \equiv 1\quad \pmod{2} \text{ and } a \equiv 1\quad \pmod{2},\\ (a+c)(1+(b_j+d_j)c) &\text{if } c \equiv 1\quad \pmod{2} \text{ and } a \equiv 0\quad \pmod{2},\\ a(1-ab_j cd_j) &\text{if } c \equiv 2\quad \pmod{4},\\ a(1-b_jc) &\text{if } c \equiv 0\quad \pmod{4}. \end{cases} \end{align*}$$

We prove the case where $c \equiv 1\ \pmod {2}$ and $a \equiv 1\ \pmod {2}$ , leaving the others to the reader. We have

$$ \begin{align*} \rho"(\gamma)_{\alpha \beta} &= \frac{1}{2} \sum_{\substack{0 \le j \le 3 \\ d_j \equiv 1\ (2)}} \left(\frac{c}{d_j}\right) \epsilon_{d_j}^{-1} \mathbf{e} \left(\frac{-ab_j}{4}\right)^\alpha \mathbf{e} \left(\frac{-j}{4}\right)^\beta = \frac{i^{-ab \alpha}}{2} \left(\frac{d}{c}\right) \sum_{\substack{0 \le j \le 3 \\ d_j \equiv 1\ (2)}} (-1)^{\frac{(c-1)(d_j-1)}{4}} \epsilon_{d_j}^{-1} i^{-j(\alpha+\beta)}. \end{align*} $$

Since the value of the sum depends only on $c, d\ \pmod {4}$ , a direct calculation yields

$$\begin{align*}\rho"(\gamma)_{\alpha \beta} = \frac{i^{-ab \alpha}}{2} \left(\frac{d}{c}\right) \frac{2\epsilon_c}{1+i} \times \begin{cases} 1 &\text{if } (\alpha, \beta) = (0,0),\\ i^{cd} &\text{if } (\alpha, \beta) = (0,1), (1,0),\\ (-1)^{d-1} &\text{if } (\alpha, \beta) = (1,1). \end{cases} \end{align*}$$

Combining simple calculations with Proposition 3.1, one obtains $\rho (\widetilde {\gamma })_{\alpha \beta }$ .

4.3 The Maass–Poincaré series

Using the two previous subsections, we now construct the Maass–Poincaré series. We let and . Assume that $k \in \frac {1}{2} \mathbb {Z}$ satisfies $2k \equiv 3\ \pmod {4}$ . For $\alpha \in \{0, 1\}$ and $m \equiv -\alpha\ \pmod {4}$ , we define the Maass–Poincaré series of weight k with respect to $\rho ^*$ by

(4.9)

This series converges absolutely and uniformly on compact subsets in $\operatorname {\mathrm {Re}}(s)> 1$ [Reference Bruinier6, p.29], and we note that is invariant under $|_{k,\rho ^*} (\gamma , \phi )$ for any $(\gamma , \phi ) \in \widetilde {\Gamma }_\infty $ as $2k \equiv 3\ \pmod {4}$ .

The Fourier expansions of the functions involve the Bessel functions (see [Reference Magnus, Oberhettinger and Soni17, Ch. 3] and [Reference Whittaker and Watson30, Ch. 17])

Proposition 4.4. For $\operatorname {\mathrm {Re}}(s)> 1$ , we have

where

$$ \begin{align*} b_{m,k}^{(\beta)}(n, s) &= 2\pi i^{-k} \sum_{c> 0} \left(1+\left(\frac{4}{c}\right)\right) \frac{K_{3/2}(m,n,4c)}{4c}\\ &\qquad \qquad \times \begin{cases} \displaystyle{|mn|^{\frac{1-k}{2}} J_{2s-1} \left(\frac{\pi \sqrt{|mn|}}{c}\right)} &\text{if } mn > 0,\\ \displaystyle{|mn|^{\frac{1-k}{2}} I_{2s-1} \left(\frac{\pi \sqrt{|mn|}}{c}\right)} &\text{if } mn < 0,\\ \displaystyle{2^{k-1} \pi^{s+k/2-1} |m+n|^{s-k/2} (4c)^{1-2s}} &\text{if } mn=0, m+n \neq 0,\\ \displaystyle{2^{2k-2} \pi^{k-1} \Gamma(2s) (8c)^{1-2s}} &\text{if } m = n = 0. \end{cases} \end{align*} $$

Proof. Dividing the sum of the Poincaré series into the identity class and the remaining part, we have

Let $H_{k, \rho ^*}^{(\alpha , m)}(\tau , s)$ denote the sum of the second term. By following the exact same argument as in the proof of Theorem 1.9 in Bruinier’s book [Reference Bruinier6], we obtain the Fourier expansion,

$$\begin{align*}H_{k,\rho^*}^{(\alpha, m)}(\tau,s) = \sum_{\beta \in \{0,1\}} \sum_{n \in \mathbb{Z}} \left( \sum_{c> 0} \sum_{d \in (\mathbb{Z}/c\mathbb{Z})^\times} \rho(\widetilde{\gamma})_{\alpha \beta} \mathbf{e}\left(\frac{ma+nd}{4c}\right) I_m(n) \right) \mathbf{e} \left(\frac{nu}{4}\right) \mathfrak{e}_\beta, \end{align*}$$

where $I_m(n)$ is given by

By combining this with Proposition 4.3, we obtain Proposition 4.4.

4.4 Traces of singular moduli

The coefficients of these functions are related to traces of singular moduli, as shown in several previous works (for example, see [Reference Bringmann and Ono5, Reference Bruinier, Jenkins and Ono8, Reference Duke, Imamoḡlu and Tóth12]). To make this precise, we consider weight 3/2 modular forms h on $\Gamma _0(4)$ satisfying

(4.10)

We define for $i \in \{0, 1\}$ , and then we have that

(4.11)

is a weight 3/2 vector-valued modular form with respect to $\rho ^*$ (see [Reference Eichler and Zagier13, Section 5] and [Reference Bringmann, Folsom, Ono and Rolen4, Ch. 2]).

We relate the $g_m(\tau )$ in (2.6) and to the Maass–Poincaré expressions

(4.12)

where $\alpha \equiv n^2\ \pmod {4}$ for each n. To be precise, we have the following theorem.

Theorem 4.5. If m is a nonnegative integer, then we have

$$\begin{align*}\lim_{s \to 3/4} G_m(\tau, s) = \begin{pmatrix}g_{m,0}(\tau) \\ g_{m,1}(\tau)\end{pmatrix}. \end{align*}$$

Sketch of the Proof. This result is standard, and so we sketch the proof. We first recall facts about Niebur–Poincaré series $F_m(\tau , s)$ (see [Reference Niebur23] or [Reference Duke, Imamoḡlu and Tóth12, Section 4]), which are defined for $\operatorname {\mathrm {Re}}(s)> 1,$ and give alternative expressions for the $j_m(\tau )$ . Specifically, as described in [Reference Duke, Imamoḡlu and Tóth12, (4.10)], it is known that

$$\begin{align*}\mathop{\mathrm{Res}}_{s=1} F_0(\tau, s) = \frac{3}{\pi} \end{align*}$$

and

(4.13) $$ \begin{align} \lim_{s \to 1} F_{-m}(\tau, s) = j_m(\tau) + 24 \sigma_1(m) \quad (m> 0). \end{align} $$

For nonnegative integers m, the trace functions

have a direct connection to the coefficients of the earlier Maass–Poincaré series. Indeed, by combining the result of Duke, Imamoḡlu and Tóth in [Reference Duke, Imamoḡlu and Tóth12, Proposition 4] with our Proposition 4.4, for $\operatorname {\mathrm {Re}}(s)> 1$ , $m \ge 0$ , and $d> 0$ with $d \equiv 0, 3\ \pmod {4}$ , we obtain that

$$\begin{align*}\mathrm{Tr}_d(F_{-m}(\cdot, s)) = \begin{cases} \displaystyle{- d^{1/2} \sum_{n|m} n b_{-n^2, 3/2}^{(\beta)} \left(d, \frac{s}{2} + \frac{1}{4}\right)} &\text{if } m> 0,\\ \displaystyle{-2^{s-2} \pi^{-s/2-1} d^{1/2} \zeta(s) b_{0, 3/2}^{(\beta)} \left(d, \frac{s}{2} + \frac{1}{4}\right)} &\text{if } m = 0. \end{cases} \end{align*}$$

Therefore, (4.13) implies that

(4.14) $$ \begin{align} \mathbf{t}_m(d) = \lim_{s \to 3/4} \begin{cases} \displaystyle{- d^{1/2} \sum_{n|m} n b_{-n^2, 3/2}^{(\beta)} (d, s) + \frac{4d^{1/2}}{\sqrt{\pi}} \sigma_1(m) b_{0, 3/2}^{(\beta)}(d,s)} &\text{if } m> 0,\\ \displaystyle{- \frac{d^{1/2}}{6 \sqrt{\pi}} b_{0, 3/2}^{(\beta)} (d, s)} &\text{if } m = 0. \end{cases} \end{align} $$

By applying (4.3) and (4.4), we thereby conclude the proof of the theorem.

5.1 Jacobi forms and the modified heat operator

For a function $\varphi : \mathbb {H} \times \mathbb {C} \to \mathbb {C}$ , $\gamma \in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ , and positive integers $k, m \in \mathbb {Z}_{>0}$ , we define the slash operator

and the weighted heat operator

where . Then, we have

(5.1) $$ \begin{align} L_{k,m} (\varphi|_{k,m} \gamma) = (L_{k,m} \varphi)|_{k+2,m} \gamma \end{align} $$

for any $\gamma \in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ (see [Reference Eichler and Zagier13, (11) in Section 3]). For simplicity, we put .

Lemma 5.1. For a Poincaré series defined by

$$\begin{align*}G(\tau) = \sum_{(\gamma, \phi) \in \widetilde{\Gamma}_\infty \backslash \operatorname{\mathrm{Mp}}_2(\mathbb{Z})} \begin{pmatrix}\psi_0(\tau) \\ \psi_1(\tau)\end{pmatrix} \bigg|_{3/2, \rho^*} (\gamma, \phi), \end{align*}$$

with test functions $\psi _0, \psi _1: \mathbb {H} \to \mathbb {C}$ , we have

$$\begin{align*}{}^t \Theta(\tau, z) G(\tau) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} (\theta_0(\tau, z) \psi_0(\tau) + \theta_1(\tau, z) \psi_1(\tau)) |_{2,1} \gamma. \end{align*}$$

Proof. By a direct calculation with (3.4), we have

$$ \begin{align*} {}^t \Theta(\tau, z) G(\tau) = \sum_{(\gamma, \phi) \in \widetilde{\Gamma}_\infty \backslash \operatorname{\mathrm{Mp}}_2(\mathbb{Z})} \phi(\tau)^{-4} \mathbf{e} \left(\frac{-cz^2}{c\tau+d}\right) {}^t \Theta\left(\gamma \tau, \frac{z}{c\tau+d}\right) {}^t \rho((\gamma, \phi))^{-1} \rho^*((\gamma, \phi))^{-1} \begin{pmatrix}\psi_0(\gamma\tau) \\ \psi_1(\gamma \tau)\end{pmatrix}. \end{align*} $$

Since ${}^t \rho ((\gamma , \phi ))^{-1} \rho ^*((\gamma , \phi ))^{-1} = I$ and $\phi (\tau )^{-4} = (c\tau +d)^{-2}$ , we obtain the result.

We require the following proposition for the $p_k(r,n)$ in the Eichler–Selberg trace formula.

Proposition 5.2. For $\nu , l \in \mathbb {Z}_{\ge 0}$ and $r \in \mathbb {Z}$ , we define the differential operator by

(5.2)

Then, for a function $: \mathbb {H} \to \mathbb {C}$ , we have the Taylor expansion

$$\begin{align*}L_{2\nu} \circ \cdots \circ L_2 f(\tau)(\zeta^r + \zeta^{-r}) = 2 \sum_{l=0}^\infty p_{2\nu+2}(r,D,l) f(\tau) \frac{(2\pi irz)^{2l}}{(2l)!}. \end{align*}$$

In particular, letting $p_k(r,n)$ as in (1.4), we have that $p_{2\nu +2}(r,D,0) = p_{2\nu +2}(r,D)$ and

$$\begin{align*}\lim_{z \to 0} L_{2\nu} \circ \cdots \circ L_2 f(\tau)(\zeta^r + \zeta^{-r}) = 2 p_{2\nu+2}(r,D) f(\tau). \end{align*}$$

Proof. We check that the Taylor coefficients of $L_{2\nu } \circ \cdots \circ L_2 f(\tau ) (\zeta ^r + \zeta ^{-r})$ and the sequence (5.2) satisfy the same recursion. The claim is clear for $\nu = 0$ . For $\nu> 0$ , let

Then $S_{\nu , l, j}$ satisfies the recursion

$$\begin{align*}S_{\nu, l, j} = -D S_{\nu-1, l, j-1} + \frac{r^2}{4} \left(1 + \frac{4\nu-1}{2l+1}\right) S_{\nu-1, l+1, j}, \end{align*}$$

for $\nu \ge 1$ and $0 \le j \le \nu $ with $S_{\nu , l, -1} = 0$ , which implies that

$$\begin{align*}p_{2\nu+2}(r,D,l) = -D p_{2\nu}(r,D,l) + \frac{r^2}{4} \left(1 + \frac{4\nu-1}{2l+1}\right) p_{2\nu} (r,D,l+1). \end{align*}$$

One can check that the Taylor coefficients also satisfy this recursion.

We use this proposition to understand the combinatorial properties of the Rankin–Cohen bracket operators, which is a slight generalization of [Reference Eichler and Zagier13, Theorem 5.5].

Proposition 5.3. Let $\nu \ge 0$ be a nonnegative integer. For a modular form h of weight $3/2$ on $\Gamma _0(4)$ of the form (4.10), we have

Proof. By definition, we have

A direct calculation implies that

and

$$ \begin{align*} \sum_{\substack{r, s \ge 0\\ r+s = \nu}} (-1)^r \frac{\Gamma(3/2+\nu) \Gamma(1/2+\nu)}{s! \Gamma(3/2+r) r! \Gamma(1/2+s)} m^{2s} (4D-m^2)^r = \binom{2\nu}{\nu} p_{2\nu+2}(m,D). \end{align*} $$

The last equation immediately follows from Proposition 5.2.

For each $n \ge 0$ and $\nu \ge 0$ , we define

(5.3)

Combining Theorem 4.5 and Lemma 5.3, for $m\geq 1,$ we obtain the following key expressions:

(5.4) $$ \begin{align} \mathcal{G}_{m,\nu}(\tau) = -\frac{1}{2\binom{2\nu}{\nu}} \cdot [g_m, \theta]_\nu |U_4 = -\frac{1}{2} \lim_{s \to 3/4} \left( -\frac{\sqrt{\pi}}{2} \sum_{n \mid m} n \Phi_{n, \nu}(\tau, s) + 2\sigma_1(m) \Phi_{0,\nu}(\tau, s)\right). \end{align} $$

The order of limits of s and z is interchanged, which is justified by the Remark at the end of Section 4.4.

5.2 The Selberg–Poincaré series

To prove Theorem 1.4 using (5.4), we must calculate $\Phi _{n,\nu }(\tau , s)$ and $\langle \Phi _{n, \nu }(\cdot , s), f\rangle $ at $s = 3/4$ for Hecke eigenforms f. To this end, we use Selberg’s generalization [Reference Selberg27] of the Poincaré series in (1.7). For integers $k\ge 2$ and $m\in \mathbb {Z},$ they are defined by

(5.5)

This series converges absolutely and uniformly on compact subsets for $\operatorname {\mathrm {Re}}(s)> 1-k/2$ and admits meromorphic continuation. In particular, it is known that $P_{k,m}(\tau , s)$ is holomorphic at $s = 1-k/2$ . This fact follows from comparing it with the Maass–Poincaré series defined by

Indeed, from (4.5), we have

Thus, for $\operatorname {\mathrm {Re}}(s)> -k/2$ , the poles of these two types of Poincaré series agree. However, the Fourier expansion of the Maass–Poincaré series (see [Reference Jeon, Kang and Kim15, Theorem 3.2]) and the Weil bound for the Kloosterman sums imply its holomorphy at $s = 1-k/2$ .

The next lemma describes the Petersson inner product of cusp forms with these series.

Lemma 5.4. For $f \in S_k$ and $m> 0$ , we have

5.3 The case of $n=0$

Here, we calculate $\langle \Phi _{0, \nu }(\cdot , s), f\rangle $ at $s = 3/4$ for a normalized Hecke eigenform f. To this end, we decompose $\Phi _{0,\nu }(\tau ,s)$ in terms of the Selberg–Poincaré series.

Proposition 5.5. We have that

$$\begin{align*}\Phi_{0,\nu}(\tau, s) = 4^{-s+3/4} \sum_{0 \le l \le \nu} \frac{(s-3/4)^{\underline{l}}}{(4\pi)^l} \binom{2\nu+1}{2l+1} \sum_{r \in \mathbb{Z}} r^{2\nu-2l} P_{2\nu+2, r^2} \left(\tau, s-\frac{3}{4}-l\right). \end{align*}$$

Proof. By applying (5.1), Lemma 5.1 and Proposition 5.2,

The summand is calculated as

Then the claim follows from the Leibniz rule, where , and the fact that

$$\begin{align*}\sum_{l \le j \le \nu} (-1)^j 2^{2(\nu-j)} \binom{2\nu-j}{j} \binom{j}{l} = (-1)^l \binom{2\nu+1}{2l+1}. \end{align*}$$

The next result provides a formula for the Petersson norm of a cusp form f.

Theorem 5.6. For a normalized Hecke eigenform $f \in S_{2\nu +2}$ , we have

$$\begin{align*}\lim_{s \to 3/4} \langle \Phi_{0,\nu} (\cdot, s), f \rangle = 24 \|f\|^2. \end{align*}$$

Proof. First, we note that the Fourier coefficients of a normalized Hecke eigenform are real. By Lemma 5.4 and Proposition 5.5, we find that

$$ \begin{align*} \langle \Phi_{0,\nu}(\cdot, s), f \rangle &= 4^{-s+3/4} \sum_{0 \le l \le \nu} \frac{(s-3/4)^{\underline{l}}}{(4\pi)^l} \frac{\Gamma(2\nu+s+1/4-l)}{(4\pi)^{2\nu+s+1/4-l}} \binom{2\nu+1}{2l+1} \cdot 2 \sum_{r =1}^\infty \frac{c_f(r^2)}{r^{2s+2\nu+1/2}}. \end{align*} $$

As in [Reference Cohen and Strömberg10, Lemma 11.12.6], let

for $f \in S_{2\nu +2}$ . Then, it is known that $B(f,s)$ admits the meromorphic continuation to the whole $\mathbb {C}$ -plane, and $L(\mathrm {Sym}^2(f), s)$ has no poles (see [Reference Cohen and Strömberg10, Remark 11.12.8]). In particular, $B(f, 2s+2\nu +1/2)$ has no pole at $s = 3/4$ . Therefore, by [Reference Cohen and Strömberg10, Corollary 11.12.7], we have

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi_{0,\nu}(\cdot, s), f\rangle &= \frac{\Gamma(2\nu+1)}{(4\pi)^{2\nu+1}} (2\nu+1) 2B(f, 2\nu+2)\\ &= \frac{2(2\nu+1)!}{(4\pi)^{2\nu+1}} \frac{6}{\pi^2} \frac{\pi}{2} \frac{(4\pi)^{2\nu+2}}{(2\nu+1)!} \langle f, f \rangle\\ &=24 \|f\|^2.\\[-47pt] \end{align*} $$

5.4 The cases of $n> 0$

We turn to the case of positive $n.$ Again, we first decompose $\Phi _{n,\nu }(\tau ,s).$

Proposition 5.7. For $n> 0$ , we have

$$ \begin{align*} \Phi_{n, \nu}(\tau, s) &= \frac{1}{\Gamma(2s)} \sum_{\substack{r \in \mathbb{Z} \\ r \equiv n\ (2)}} \sum_{\substack{i_1, i_2 \ge 0 \\ i_1 + i_2 \le \nu}} \frac{(-1)^{i_1}}{i_1! i_2!} \left(\frac{n^2}{4} \right)^{i_1+i_2} Q_{\nu, i_1+i_2} (n,r) \frac{(s-3/4)^{\underline{i_1}} (s-3/4)^{\overline{i_2}}}{(2s)^{\overline{i_2}}}\widetilde{P}_{n,r}^{i_1, i_2} (\tau, s), \end{align*} $$

where we let

Proof. Arguing as above, by applying (5.1), Lemma 5.1 and Proposition 5.2, we obtain

The summand is calculated as

Similar to the case of $n=0$ , direct calculation utilizing yields

For the second term, by Lemma 4.1, we find that

The claim follows by combining these results.

We split the sum defining $\Phi _{n, \nu }(\tau , s)$ into $\Phi _{n, \nu }^+(\tau , s)$ and $\Phi _{n, \nu }^-(\tau , s),$ based on the inequalities $r^2> n^2$ or $r^2 \le n^2.$ We consider them as $s \to 3/4$ . By (4.5), the summand of the Poincaré series $\widetilde {P}_{n,r}^{i_1, i_2} (\tau , s)$ satisfies

as . Therefore, for $\operatorname {\mathrm {Re}}(s)> -\nu +i_1 + 3/4$ , the Poincaré series is holomorphic (in s). In particular, $\widetilde {P}_{n,r}^{i_1, i_2} (\tau , s)$ is holomorphic at $s = 3/4$ for $0 \le i_1 < \nu $ . Regarding the case of $i_1 = \nu $ , by a similar argument as in Section 5.2 – that is, by comparing it with the Selberg–Poincaré series or the Maass–Poincaré series – we see that it is also holomorphic at $s=3/4$ . Therefore, we have

$$ \begin{align*} \lim_{s \to 3/4} \Phi_{n, \nu}^-(\tau, s) &= \frac{1}{\Gamma(3/2)} \sum_{\substack{r \in \mathbb{Z} \\ r \equiv n\ (2)\\r^2 \le n^2}} Q_{\nu,0} (n,r) \widetilde{P}_{n,r}^{0,0}(\tau, 3/4). \end{align*} $$

Since $Q_{\nu ,0}(n,r) = p_{2\nu +2}(r, (r^2-n^2)/4)$ and $\widetilde {P}_{n,r}^{0,0}(\tau , 3/4) = P_{2\nu +2, \frac {r^2-n^2}{4}}(\tau )$ , by (2.8), we have

(5.6) $$ \begin{align} \lim_{s \to 3/4} \Phi_{n, \nu}^-(\tau, s) = \frac{4}{n \sqrt{\pi}} \sum_{0 < r \le n} r^{2\nu+1} P_{2\nu+2, -r(n-r)}(\tau). \end{align} $$

As a counterpart to Theorem 5.6, the Petersson inner product of $\Phi _{n, \nu }^+(\tau , s)$ with a Hecke eigenform is expressed in terms of the symmetrized shifted convolution L-functions.

Theorem 5.8. For a normalized Hecke eigenform $f \in S_{2\nu +2}$ , we have

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi_{n,\nu}^+(\cdot, s), f \rangle = \frac{4}{n \sqrt{\pi}} \frac{\Gamma (2\nu + 1)}{(4\pi)^{2\nu + 1}} \sum_{d \mid n} \mu(d) \widehat{L}(f, n/d; 2\nu+1). \end{align*} $$

Proof. By Proposition 5.7, we have

$$ \begin{align*} &\langle \Phi_{n, \nu}^+(\cdot, s), f \rangle\\ &= \frac{1}{\Gamma(2s)} \sum_{\substack{r \in \mathbb{Z} \\ r \equiv n\ (2) \\ r^2> n^2}} \sum_{\substack{i_1, i_2 \ge 0 \\ i_1 + i_2 \le \nu}} \frac{(-1)^{i_1}}{i_1! i_2!} \left(\frac{n^2}{4} \right)^{i_1+i_2} Q_{\nu, i_1+i_2} (n,r) \frac{(s-3/4)^{\underline{i_1}} (s-3/4)^{\overline{i_2}}}{(2s)^{\overline{i_2}}} \langle \widetilde{P}_{n,r}^{i_1, i_2} (\cdot, s), f \rangle. \end{align*} $$

The unfolding argument, combined with Lemma 4.2, gives

By changing variables $r = 2m+n$ for $r> n$ and $r = -2m-n$ for $r < -n$ , we have

$$ \begin{align*} &\langle \Phi_{n,\nu}^+(\cdot, s), f\rangle\\ &= \frac{2}{\Gamma(2s)} \sum_{\substack{i_1, i_2 \ge 0 \\ i_1 + i_2 \le \nu}} \frac{(-1)^{i_1}}{i_1! i_2!} \left(\frac{n^2}{4} \right)^{i_1+i_2} \frac{(s-3/4)^{\underline{i_1}} (s-3/4)^{\overline{i_2}}}{(2s)^{\overline{i_2}}} (\pi n^2)^{s -\frac{3}{4}-i_1} \Gamma \left(s + 2\nu + \frac{1}{4} - i_1 \right)\\ &\times \sum_{m=1}^\infty \frac{Q_{\nu, i_1+i_2} (n,2m+n) c_f (m(m+n))}{(\pi (2m+n)^2)^{s + 2\nu + \frac{1}{4} - i_1}} {}_2 F_1 \left(s+\frac{3}{4}, s + 2\nu + \frac{1}{4} - i_1; 2s+i_2; \frac{n^2}{(2m+n)^2}\right), \end{align*} $$

where we note that $Q_{\nu ,i}(n,-r) = Q_{\nu ,i}(n,r)$ holds. For a normalized Hecke eigenform $f \in S_{2\nu +2}$ , since

$$\begin{align*}c_f(m(m+n)) = \sum_{d \mid (m, m+n)} \mu(d) d^{2\nu+1} c_f\left(\frac{m}{d}\right) c_f\left(\frac{m+n}{d}\right), \end{align*}$$

the last sum becomes

$$ \begin{align*} &\sum_{d \mid n} \mu(d) d^{2\nu+1} \sum_{m=1}^\infty \frac{Q_{\nu, i_1+i_2} (n,2dm+n) c_f(m) c_f(m+n/d)}{(\pi (2dm+n)^2)^{s + 2\nu + \frac{1}{4} - i_1}}\\ &\quad \times {}_2 F_1 \left(s+\frac{3}{4}, s + 2\nu + \frac{1}{4} - i_1; 2s+i_2; \frac{n^2}{(2dm+n)^2}\right).\end{align*} $$

Then, this Dirichlet series is holomorphic at $s = 3/4$ . Indeed, since $Q_{\nu , i_1+i_2} (n,2dm+n)$ has degree $2(\nu -i_1-i_2)$ in m, it suffices to show that

$$\begin{align*}\sum_{m=1}^\infty \frac{c_f(m) c_f(m+n/d)}{m^{2(s+\nu+1/4+i_2)}} \end{align*}$$

(conditionally) converges at $s = 3/4$ . This can be seen by partial summation using the estimate

$$\begin{align*}\sum_{1 \le m \le x} c_f(m) c_f(m+n/d) \ll x^{2\nu+2-\delta}, \end{align*}$$

with some $\delta> 0$ , (see [Reference Blomer2, Corollary 1.4]). Therefore, all terms corresponding to nonzero $(i_1, i_2)$ vanish as $s \to 3/4$ , and we obtain

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi_{n,\nu}^+(\cdot, s), f\rangle &= \frac{4}{\sqrt{\pi}} \Gamma(2\nu+1) \sum_{d \mid n} \mu(d) d^{2\nu+1} \\ &\quad \times \sum_{m=1}^\infty \frac{Q_{\nu, 0} (n,2dm+n) c_f(m) c_f(m+n/d)}{(\pi (2dm+n)^2)^{2\nu + 1}} {}_2 F_1 \left(\frac{3}{2}, 2\nu + 1; \frac{3}{2}; \frac{n^2}{(2dm+n)^2}\right). \end{align*} $$

Since we have

$$\begin{align*}\frac{1}{r^{2(2\nu+1)}} {}_2 F_1 \left(\frac{3}{2}, 2\nu + 1; \frac{3}{2}; \frac{n^2}{r^2}\right) = \frac{1}{(r^2-n^2)^{2\nu+1}} \end{align*}$$

and $Q_{\nu ,0}(n,r) = p_{2\nu +2}(r, (r^2-n^2)/4)$ with (2.8) again, the proof is complete as

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi_{n,\nu}^+(\cdot, s), f\rangle &= \frac{4}{n \sqrt{\pi}} \frac{\Gamma(2\nu+1)}{(4\pi)^{2\nu + 1}} \sum_{d \mid n} \mu(d) \sum_{m=1}^\infty c_f(m) c_f(m+n/d) \left(\frac{1}{m^{2\nu+1}} - \frac{1}{(m+n/d)^{2\nu+1}}\right). \end{align*} $$

5.5 Proof of Theorem 1.4

We apply the results from the two previous subsections. For $m> 0$ and $\nu \ge 0$ , let

As stated in (5.4), we have

$$\begin{align*}\lim_{s \to 3/4} \Phi(\tau, s) = \mathcal{G}_{m,\nu}(\tau). \end{align*}$$

However, from (5.6), the minus part

For the plus part, by Theorem 5.6 and Theorem 5.8 and the Möbius inversion formula, we have

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi^+(\cdot, s), f \rangle &= \sum_{n \mid m} \frac{\Gamma (2\nu + 1)}{(4\pi)^{2\nu + 1}} \sum_{d \mid n} \mu(d) \widehat{L}(f, n/d; 2\nu+1) - 24\sigma_1(m) \|f\|^2\\ &= \frac{\Gamma (2\nu + 1)}{(4\pi)^{2\nu + 1}} \widehat{L}(f, m; 2\nu+1) - 24 \sigma_1(m) \|f\|^2. \end{align*} $$

Combining these facts, we are pleased to obtain the conclusion of the theorem

$$ \begin{align*} \lim_{s \to 3/4} \Phi(\tau, s) &= \sum_{n \mid m} \sum_{0 < r \le n} r^{2\nu+1} P_{2\nu+2, -r(n-r)}(\tau) - \sum_{j=1}^{d_{2\nu+2}} \left(24 \sigma_1(m) - \frac{\Gamma (2\nu + 1)}{(4\pi)^{2\nu + 1}} \frac{\widehat{L}(f, m; 2\nu+1)}{\|f_j\|^2} \right) f_j. \end{align*} $$