2.1 Manipulations of character sums

The orthogonality relation asserts that

(2.1) $$ \begin{align} \sum_{a\! \pmod q} \chi(a) = \begin{cases} \varphi(q) & \text{if } \chi = \chi_{0},\\ 0 & \text{otherwise}, \end{cases} \qquad \sum_{\chi \pmod{q} } \chi(a) = \begin{cases} \varphi(q) & \text{if } a \equiv 1 \pmod q,\\ 0 & \text{otherwise}. \end{cases} \\[-15pt]\nonumber\end{align} $$

For any Dirichlet character $\chi $ modulo q, let

(2.2) $$ \begin{align} \tau(\chi, h) = \sum_{b\! \pmod q} \chi(b) e \left(\frac{bh}{q} \right) \\[-15pt]\nonumber\end{align} $$

denote the Gauß sum associated to characters on residue classes modulo q. One writes $\tau (\chi ) = \tau (\chi , 1)$ as usual. Multiplying equation (2.2) by $\overline {\chi }(a)$ and summing over $\chi $ , we derive via orthogonality equation (2.1) that

(2.3) $$ \begin{align} e \left(\frac{ah}{q} \right) = \frac{1}{\varphi(q)} \sum_{\chi \pmod{q} } \overline{\chi}(a) \tau(\chi, h) \quad \text{if} \quad (a, q) = 1.\\[-15pt]\nonumber \end{align} $$

This is the Fourier expansion of additive characters in terms of the multiplicative ones. One can evaluate Gauß sums for general Dirichlet characters.

Lemma 2.1. Let $\chi $ be a nontrivial Dirichlet character modulo q induced by the primitive character $\chi ^{\ast }$ modulo $q^{\ast }$ . For an integer $n \geqslant 1$ , we have that

$$ \begin{align*} \tau(\chi, n) = \tau(\chi^{\ast}) \sum_{d \mid (n, q/q^{\ast})} d \overline{\chi^{\ast}} \left(\frac{n}{d} \right) \chi^{\ast} \left(\frac{q}{dq^{\ast}} \right) \mu \left(\frac{q}{dq^{\ast}} \right).\\[-15pt] \end{align*} $$

We indicate by $r_{c}(n)$ and $S(m, n; c)$ the Ramanujan sum and Kloosterman sum, respectively, as follows:

(2.4)

where the asterisk means that the summation is restricted to a reduced system of residues. We have the Weil bound

(2.5) $$ \begin{align} |S(m, n; c)| \leqslant (m, n, c)^{1/2} c^{1/2} \tau(c).\\[-15pt]\nonumber \end{align} $$

This gives the best possible bound for individual Kloosterman sums, whereas we are apt to make use of the Kuznetsov formula (Theorem 2.11) to obtain additional savings from the sum over the moduli c. Indeed, various arithmetic problems can be transformed into bounding sums of Kloosterman sums. The twisted multiplicativity for Kloosterman sums is occasionally exploited. We also use the Jacobi sum

$$ \begin{align*} J(\chi, \psi) = \sum_{a \pmod{q} } \chi(a) \psi(1-a).\\[-15pt] \end{align*} $$

In particular, when $\chi $ and $\psi $ are of the same modulus and $\chi \psi $ is primitive, the relation between the Gauß sum and Jacobi sum is illustrated as (see [Reference Iwaniec and Kowalski38, (3.18)])

(2.6) $$ \begin{align} J(\chi, \psi) = \frac{\tau(\chi) \tau(\psi)}{\tau(\chi \psi)}. \end{align} $$

In general, we can establish the following lemma:

Lemma 2.2. Assume that q is prime. Suppose that $\psi _{1}, \psi _{2}$ are primitive characters modulo q satisfying $\psi _{1} \ne \overline {\psi _{2}}$ , and let $a, b, c, d \in \mathbb {Z}$ with $(a, c, q) = 1$ . Then

(2.7) $$ \begin{align} \sum_{t \pmod{q} } \psi_{1}(at+b) \psi_{2}(ct+d) = \overline{\psi_{1}}(c) \overline{\psi_{2}}(a) \psi_{1} \psi_{2}(ad-bc) \frac{\tau(\psi_{1}) \tau(\overline{\psi_{1} \psi_{2}})}{\tau(\overline{\psi_{2}})}. \end{align} $$

Moreover, when $\psi _{1} = \overline {\psi _{2}} = \psi $ , we have that

(2.8) $$ \begin{align} \sum_{t \pmod{q} } \psi(at+b) \overline{\psi}(ct+d) = \psi(a) \overline{\psi}(c) r_{q}(ad-bc). \end{align} $$

Proof. The sum vanishes unless $(a, q) = (c, q) = 1$ as in [Reference Petrow and Young64, Page 455]. We assume $(a, q) = (c, q) = 1$ in what follows. To prove the first assertion, we initially expand the character $\psi _{1}$ into exponentials:

$$ \begin{align*} \psi_{1}(at+b) = \frac{1}{\tau(\overline{\psi_{1}})} \sum_{x \pmod{q} } \overline{\psi_{1}}(x) e_{q}((at+b)x). \end{align*} $$

We would like to calculate the t-sum to reduce the problem to the manipulation of the x-sum. Then

$$ \begin{align*} \sum_{t \pmod{q} } \psi_{2}(ct+d) e_{q}(atx) = \sum_{t \pmod{q} } \psi_{2}(t) e_{q}(a \overline{c}(t-d)x) = \overline{\psi_{2}}(ax) \psi_{2}(c) e_{q}(-a \overline{c} dx) \tau(\psi_{2}). \end{align*} $$

On the other hand, one has

$$ \begin{align*} \sum_{x \pmod{q} } \overline{\psi_{1} \psi_{2}}(x) e_{q}(x(b-a \overline{c} d)) = \psi_{1} \psi_{2}(ad-bc) \overline{\psi_{1} \psi_{2}}(-c) \tau(\overline{\psi_{1} \psi_{2}}). \end{align*} $$

Gathering the above identities together, we arrive at

$$ \begin{align*} \sum_{t \pmod{q} } \psi_{1}(at+b) \psi_{2}(ct+d) = \overline{\psi_{1}}(c) \overline{\psi_{2}}(a) \psi_{1} \psi_{2}(bc-ad) \frac{\tau(\psi_{2}) \tau(\overline{\psi_{1} \psi_{2}})}{\tau(\overline{\psi_{1}})}. \end{align*} $$

Applying the relations $\tau (\overline {\psi _{1}}) = \psi _{1}(-1) \tau (\psi _{1})^{-1} q$ and $\tau (\psi _{2}) = \psi _{2}(-1) \tau (\overline {\psi _{2}})^{-1} q$ , the desired expression follows. The second claim is the same as in [Reference Petrow and Young64, Lemma 6.3].

As mentioned in the introduction, we will encounter sums of the product of Kloosterman sums. One should resolve $\overline {\ell }$ inside the argument of the Kloosterman sum, and the following lemma is helpful when we expand $S(a \overline {\ell }, \pm n \overline {\ell }; q)$ into multiplicative characters.

Lemma 2.3. Assume that $q \geqslant 1$ and $a, b \in \mathbb {Z}$ . We write $a = a_{0} a^{\prime }$ and $b = b_{0} b^{\prime }$ , where $a_{0} b_{0} \mid q^{\infty }$ and $(a^{\prime } b^{\prime }, q) = 1$ . Then

(2.9) $$ \begin{align} S(a, b; q) = \frac{1}{\varphi(q)} \sum_{\psi \pmod{q} } \tau(\psi, a_{0}) \tau(\psi, b_{0}) \overline{\psi}(a^{\prime} b^{\prime}), \end{align} $$

where the sum runs over all Dirichlet characters modulo q and $\varphi (q)$ is Euler’s totient function.

Proof. We exploit equation (2.3) so that the Kloosterman sum $S(a, b; q)$ equals

$$ \begin{align*} \sideset{}{^{\ast}} \sum_{d\! \pmod{q} } e \left(\frac{a_{0} d+b_{0} a^{\prime} b^{\prime} \overline{d}}{q} \right) & = \frac{1}{\varphi(q)} \ \sideset{}{^{\ast}} \sum_{d\! \pmod{q} } e \left(\frac{a_{0} d}{q} \right) \sum_{\psi \pmod{q} } \overline{\psi}(a^{\prime} b^{\prime} \overline{d}) \tau(\psi, b_{0})\\ & = \frac{1}{\varphi(q)} \sum_{\psi \pmod{q} } \tau(\psi, a_{0}) \tau(\psi, b_{0}) \overline{\psi}(a^{\prime} b^{\prime}). \end{align*} $$

This finishes the proof of the lemma.

Direct corollaries of Lemma 2.3 include

Corollary 2.4. Suppose that $q \geqslant 1$ and $a, b \in \mathbb {Z}$ . For $(ab, q) = 1$ , we then have

$$ \begin{align*} S(a, b; q) = \frac{1}{\varphi(q)} \sum_{\psi \pmod{q} } \tau(\psi)^{2} \overline{\psi}(ab). \end{align*} $$

This idea of separation of variables traces back to celebrated work of Blomer–Milićević in [Reference Blomer and Milićević15]. The number of the Gauß sums is the crux in Lemma 2.3 as the nth power determines the hyper-Kloosterman sum of n variables. A variation of Lemma 2.3 was employed by Petrow–Young [Reference Petrow and Young64, Lemma 8.8], where they handled the hyper-Kloosterman sum $\mathrm {Kl}_{3}(x, y, z; q)$ .

2.2 Double Estermann zeta function

We introduce the divisor function

$$ \begin{align*} \sigma_{w}(m) = \sum_{d \mid m} d^{w}. \end{align*} $$

As far as we know, the proof of the Hecke relation for $\sigma _{w}(m)$ has not appeared in the antecedent literature.

Lemma 2.5. The divisor function satisfies the following multiplicativity relation:

(2.10) $$ \begin{align} \sigma_{w}(mn) = \sum_{c \mid (m, n)} \mu(c) c^{w} \sigma_{w} \left(\frac{m}{c} \right) \sigma_{w} \left(\frac{n}{c} \right). \end{align} $$

Proof. If $m = p_{1}^{k_{1}} p_{2}^{k_{2}} \cdots p_{r}^{k_{r}}$ , $n = p_{1}^{\ell _{1}} p_{2}^{\ell _{2}} \cdots p_{r}^{\ell _{r}}$ and such a formula is shown for powers of primes, then

(2.11) $$ \begin{align} \begin{split} \sigma_{w}(mn) &= \sigma_{w}(p_{1}^{k_{1}+\ell_{1}}) \sigma_{w}(p_{2}^{k_{2}+\ell_{2}}) \cdots \sigma_{w}(p_{r}^{k_{r}+\ell_{r}})\\ & = \prod_{i = 1}^{r} \sum_{j = 0}^{\min(k_{i}, \ell_{i})} \mu(p_{i}^{j}) p_{i}^{jw} \sigma_{w}(p_{i}^{k_{i}-j}) \sigma_{w}(p_{i}^{\ell_{i}-j})\\ & = \sum_{c \mid (m, n)} \mu(c) c^{w} \sigma_{w} \left(\frac{m}{c} \right) \sigma_{w} \left(\frac{n}{c} \right). \end{split} \end{align} $$

By multiplicativity, it suffices to prove the result for $m = p^{k}$ , $n = p^{\ell }$ . The formula is obvious if $\min (k, \ell ) = 0$ , so we assume both the variables are at least $1$ . Setting $X = p^{w}$ , we find that the right-hand side of equation (2.11) reads

$$ \begin{align*} \text{RHS} &= \sigma_{w}(p^{k}) \sigma_{w}(p^{\ell})-p^{w} \sigma_{w}(p^{k-1}) \sigma_{w}(p^{\ell-1})\\ & = (1+X+\cdots+X^{k})(1+X+\cdots+X^{\ell})-X(1+X+\cdots+X^{k-1})(1+X+\cdots+X^{\ell-1})\\ & = \frac{X^{k+\ell+1}-1}{X-1} = (1+p^{w}+p^{2w}+\cdots+p^{(k+\ell)w}) = \sigma_{w}(p^{k+\ell}). \end{align*} $$

This establishes Lemma 2.5.

For positive integers $h, \ell $ with $(h, \ell ) = 1$ and $\Re (s)> 1$ , we define the double Estermann zeta function as

(2.12) $$ \begin{align} D_{2} \left(s, \lambda; \frac{h}{\ell} \right) = \sum_{n = 1}^{\infty} \sigma_{\lambda}(n) e \left(\frac{nh}{\ell} \right) n^{-s}. \end{align} $$

Most of the analytic properties of $D_{2}(s, \lambda; h/\ell )$ follow from the identity

$$ \begin{align*} D_{2} \left(s, \lambda; \frac{h}{\ell} \right) = \ell^{\lambda-2s} \sum_{a, b\! \pmod{\ell}} e \left(\frac{abh}{\ell} \right) \zeta \left(s, \frac{a}{\ell} \right) \zeta \left(s-\lambda, \frac{b}{\ell} \right), \end{align*} $$

where for $\alpha \in \mathbb {R}$ and $\Re (s)> 1$ ,

is the Hurwitz zeta function. It has meromorphic continuation to the entire complex plane $\mathbb {C}$ with a simple pole at $s = 1$ of residue 1 and satisfies the functional equation

(2.13) $$ \begin{align} \zeta(s, \alpha) = \sum_{\pm} G^{\mp}(1-s) \zeta^{(\pm \alpha)}(1-s), \end{align} $$

where $G^{\pm }(s) = (2\pi )^{-s} \Gamma (s) e(\pm s/4)$ and $\zeta ^{(\alpha )}(s)$ is a meromorphic continuation of $\sum _{n \geqslant 1} e(\alpha n) n^{-s}$ . For $\alpha \in \mathbb {Q}$ , the formula in equation (2.13) is a restatement of the Poisson summation in residue classes. Hence, as a function of the single variable s, the Estermann zeta function in equation (2.12) also has meromorphic continuation to all $s \in \mathbb {C}$ with two simple poles at $s = 1$ and $s = 1+\lambda $ with respective residues $\ell ^{\lambda -1} \zeta (1-\lambda )$ and $\ell ^{-\lambda -1} \zeta (1+\lambda )$ , provided $\lambda \ne 0$ . In the case of $\lambda = 0$ , there is a double pole at $s = 1$ , and the Laurent expansion is given by

$$ \begin{align*} D_{2} \left(s, 0; \frac{h}{\ell} \right) = \frac{1/\ell}{(s-1)^{2}}+\frac{2(\gamma-\log \ell)/\ell}{(s-1)}+\cdots, \end{align*} $$

where $\gamma $ is the Euler–Mascheroni constant. The functional equation for $D_{2}(s, \lambda; h/\ell )$ reads as follows:

Theorem 2.6 ([Reference Motohashi55, Lemma 3.7])

The Estermann zeta function satisfies the functional equation

(2.14) $$ \begin{align} D_{2} \left(s, \lambda; \frac{h}{\ell} \right) &= 2(2\pi)^{2s-\lambda-2} \ell^{1+\lambda-2s} \Gamma(1-s) \Gamma(1+\lambda-s) \notag \\ & \qquad \times \left[D_{2} \left(1-s, -\lambda; \frac{\overline{h}}{\ell} \right) \cos \left(\frac{\pi \lambda}{2} \right) - D_{2} \left(1-s, -\lambda; -\frac{\overline{h}}{\ell} \right) \cos \left(\pi \left(s-\frac{\lambda}{2} \right) \right) \right], \end{align} $$

where $\overline {h}$ is the multiplicative inverse of h modulo $\ell $ : that is, $h \overline {h} \equiv 1 \pmod {\ell }$ .

The formula in equation (2.14) was first proven by Hecke and Estermann in connection with an integral representation of the Hurwitz zeta function. We already find fragments of the Kloosterman sum in equation (2.14), which incline us to apply the Kloosterman summation formula. The functional equation of the Estermann zeta function $D_{2}$ is essentially equivalent to Voronoĭ summation. Indeed, the Estermann zeta function involves a divisor function instead of Hecke eigenvalues, and this corresponds to the standard Eisenstein series.

2.3 Kloosterman sums at singular cusps

Our presentation of cusps and scaling matrices is inspired by [Reference Iwaniec37]. We restrict our attention to cusps with respect to the Hecke congruence subgroup

Let $q_{0} \mid q$ . In the following, the letter $\psi $ denotes a Dirichlet character modulo $q_{0}$ with $\kappa = (1-\psi (-1))/2$ such that $\psi (-1) = (-1)^{\kappa }$ . Then we extend $\psi $ via the identification

$$ \begin{align*} \psi \left(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = \psi(d) = \overline{\psi}(a). \end{align*} $$

The group $\Gamma $ acts transitively on $\mathbb {P}^{1}(\mathbb {Q})$ by fractional linear transformations. An element $\mathfrak {a} \in \mathbb {P}^{1}(\mathbb {Q})$ is called a cusp. Two cusps $\mathfrak {a}$ and $\mathfrak {b}$ are termed equivalent under $\Gamma $ if there exists $\gamma \in \Gamma $ satisfying $\mathfrak {a} = \gamma \mathfrak {b}$ . We write $q = rs$ with $(r, s) = 1$ and $q_{0} \mid s$ . Then we call a cusp of the form $\mathfrak {a} = 1/r$ an Atkin–Lehner cusp. The Atkin–Lehner cusps are equivalent to $\infty $ under the Atkin–Lehner operator as in [Reference Kıral and Young46, Definition 2.4]. Motohashi [Reference Motohashi57] singled out the scaling matrices corresponding to the Atkin–Lehner operators and this is crucial since the Fourier coefficients around the cusp $1/r$ are proportional to the Fourier coefficients around the cusp $\infty $ for an eigenform of the Atkin–Lehner operators.

Let $\Gamma _{\mathfrak {a}} = \{\gamma \in \Gamma : \gamma \mathfrak {a} = \mathfrak {a} \}$ be the stabiliser of the cusp $\mathfrak {a}$ in $\Gamma $ . A matrix $\sigma _{\mathfrak {a}} \in \textrm{SL}_{2}(\mathbb {R})$ , satisfying

$$ \begin{align*} \sigma_{\mathfrak{a}} \infty = \mathfrak{a} \quad \text{and} \quad \sigma_{\mathfrak{a}}^{-1} \Gamma_{\mathfrak{a}} \sigma_{\mathfrak{a}} = \Gamma_{\infty} = \left\{\pm \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}: n \in \mathbb{Z} \right\} \end{align*} $$

is called a scaling matrix for the cusp $\mathfrak {a}$ . Since the scaling matrix $\sigma _{\mathfrak {a}}$ is not uniquely determined, the choice of $\sigma _{\mathfrak {a}}$ will be important in our subsequent discussions.

Definition 2.7. Let $\mathfrak {a}, \mathfrak {b}$ be cusps and $\sigma _{\mathfrak {a}}, \sigma _{\mathfrak {b}}$ be scaling matrices. The set

$$ \begin{align*} \mathcal{C}(\mathfrak{a}, \mathfrak{b}) = \left\{{c}> 0: \begin{pmatrix} \ast & \ast \\ {c} & \ast \end{pmatrix} \in \sigma_{\mathfrak{a}}^{-1} \Gamma \sigma_{\mathfrak{b}} \right\} \end{align*} $$

is called the set of allowed moduli.

For a cusp $\mathfrak {a}$ and a scaling matrix $\sigma _{\mathfrak {a}}$ , let $u_{\mathfrak {a}}$ be such that $\sigma _{\mathfrak {a}}^{-1} u_{\mathfrak {a}} \sigma _{\mathfrak {a}} = \left (\begin {smallmatrix} 1 & 1 \\ & 1 \end {smallmatrix} \right )$ . If $\psi $ satisfies $\psi (u_{\mathfrak {a}}) = 1$ , then we say that $\mathfrak {a}$ is singular with respect to $\psi $ . It behooves one to mention the following proposition:

Proposition 2.8 ([Reference Kıral and Young46, Proposition 2.6])

Assume that $q = rs$ with $(r, s) = 1$ with $q_{0} \mid s$ , where $q_{0}$ is the modulus of $\psi $ . The two cusps $\infty $ and $1/r$ are then singular with respect to $\psi $ . We choose a scaling matrix $\sigma _{1/r}$ associated to the Atkin–Lehner cusp $1/r$ to be an Atkin–Lehner operator, namely

$$ \begin{align*} \sigma_{1/r} = \tau_{r} \nu_{s} \quad \text{with} \quad \tau_{r} = \begin{pmatrix} 1 & (s \overline{s}-1)/r \\ r & s \overline{s} \end{pmatrix}, \quad \nu_{s} = \begin{pmatrix} \sqrt{s} & \\ & 1/\sqrt{s} \end{pmatrix} \end{align*} $$

with $s \overline {s} \equiv 1 \pmod {r}$ . Then the set of allowed moduli is given by

$$ \begin{align*} \mathcal{C}(\infty, 1/r) = \{\gamma = c \sqrt{s}: c \equiv 0 \pmod{r}, \ (c, s) = 1 \}. \end{align*} $$

An important example is when $r = 1$ and $s = q$ . In this case, one has

$$ \begin{align*} \mathcal{C}(\infty, 0) = \{c \sqrt{q}: c \geqslant 1, \ (c, q) = 1 \}. \end{align*} $$

We now define Kloosterman sums with respect to a pair of cusps and general central character.

Definition 2.9. If $\mathfrak {a}, \mathfrak {b}$ are singular cusps for $\psi $ modulo $q_{0}$ , then the Kloosterman sum associated to $\mathfrak {a}$ , $\mathfrak {b}$ and $\psi $ with modulus c is defined as

(2.15) $$ \begin{align} S_{\mathfrak{ab}}(m, n; c; \psi) = \sum_{\gamma = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \Gamma_{\infty} \backslash \sigma_{\mathfrak{a}}^{-1} \Gamma \sigma_{\mathfrak{b}}/\Gamma_{\infty}} \psi(\mathrm{sgn}(c)) \overline{\psi(\sigma_{\mathfrak{a}} \gamma \sigma_{\mathfrak{b}}^{-1})} e \left(\frac{am+dn}{c} \right). \end{align} $$

The occurrence of $-I \in \Gamma _{\infty }$ indicates that the lower-left entry c is only defined up to $\pm $ sign, so that the factor $\psi (\mathrm {sgn}(c))$ accounts for this. Definition 2.9 is sensitively dependent upon the choice of $\sigma _{\mathfrak {a}}$ and $\sigma _{\mathfrak {b}}$ . Moreover, if $|c| \notin \mathcal {C}(\mathfrak {a}, \mathfrak {b})$ , then the sum appearing in equation (2.15) is empty, thus $S_{\mathfrak {ab}}(m, n; c; \psi ) = 0$ . If we stick to the case of $\mathfrak {a} = \infty $ and $\mathfrak {b} = 0$ , we know the identity (see [Reference Kıral and Young46, (2.20)])

(2.16) $$ \begin{align} S_{\infty 0}(m, n; c \sqrt{q}; \psi) = \overline{\psi}(c) S(m, n \overline{q}; c) \end{align} $$

with $(c, q) = 1$ and $q \overline {q} \equiv 1 \pmod {c}$ . This differs from the one shown in [Reference Iwaniec36, Page 58] by an additive character, which is due to a different choice of the scaling matrix. In the formula in equation (2.16), the presence of the character $\overline {\psi }(c)$ is a nice feature of the $(\infty , 0)$ cusp-pair as opposed to the $(\infty , \infty )$ cusp-pair:

(2.17) $$ \begin{align} S_{\infty \infty}(m, n; c; \psi) = S_{\psi}(m, n; c) = \sideset{}{^{\ast}} \sum_{d\! \pmod{c} } \psi(d) e \left(\frac{md+n \overline{d}}{c} \right). \end{align} $$

2.4 Normalisation and Hecke eigenvalues

In order to formulate our Kloosterman summation formula, we borrow the normalisation from [Reference Blomer, Harcos and Michel9, Reference Duke, Friedlander and Iwaniec23]. We refer the reader to [Reference Proskurin65] for an exposition in the case of general multiplier systems. Let $\Gamma \backslash \mathbb {H}$ be the modular surface, where $\Gamma = \Gamma _{0}(q)$ is the Hecke congruence subgroup and $\mathbb {H} = \{z \in \mathbb {C}: \Im (z)> 0 \}$ is the upper half-plane. There exist various self-adjoint and pairwise commuting operators acting on the space $L^{2}(\Gamma \backslash \mathbb {H})$ : the hyperbolic Laplacian

$$ \begin{align*} \Delta = -y^{2} \left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}} \right)+i\kappa y \frac{\partial}{\partial x}, \end{align*} $$

the Hecke operators (the non-archimedean counterparts of $\Delta $ )

(2.18) $$ \begin{align} (T_{n} f)(z) = \frac{1}{\sqrt{n}} \sum_{ad = n} \psi(a) \sum_{b\! \pmod{d}} f \left(\frac{az+b}{d} \right), \end{align} $$

and the reflection operator $(T_{-1} f)(z) = f(-\overline {z})$ , which flips positive and negative Fourier coefficients. As shall be explained later, we have the spectral decomposition of $L^{2}(\Gamma \backslash \mathbb {H})$ in terms of pure point spectrum, residual spectrum and continuous spectrum, namely

$$ \begin{align*} L^{2}(\Gamma \backslash \mathbb{H}) = L_{\text{cusp}}^{2}(\Gamma \backslash \mathbb{H}) \oplus L_{\text{res}}^{2}(\Gamma \backslash \mathbb{H}) \oplus L_{\text{cont}}^{2}(\Gamma \backslash \mathbb{H}). \end{align*} $$

For an integer $k> 0$ with $k \equiv \kappa \pmod {2}$ , we choose a basis $\mathcal {B}_{k}(q, \psi )$ of holomorphic cusp forms. It is taken orthonormal with respect to the weight k Petersson inner product

(2.19) $$ \begin{align} \langle h_{1}, h_{2} \rangle = \int_{\Gamma \backslash \mathbb{H}} h_{1}(z) \overline{h_{2}(z)} y^{k} \frac{dx dy}{y^{2}}, \end{align} $$

where $z = x+iy$ . We let $\mathcal {B}_{\kappa }(q, \psi )$ denote a basis of the space of Maaß cusp forms. In particular, they are eigenfunctions on $\mathbb {H}$ , are automorphic of weight $\kappa \in \{0, 1 \}$ , are square-integrable on a fundamental domain and vanish at all the cusps. Moreover, they are eigenfunctions of the $L^{2}$ -extension of the hyperbolic Laplacian $\Delta $ . For $f \in \mathcal {B}_{\kappa }(q, \psi )$ , we write $\Delta f = s(1-s)f$ with $s = 1/2+it_{f}$ and $t_{f} \in \mathbb {R} \cup i[-1/2, 1/2]$ . One may choose the basis $\mathcal {B}_{\kappa }(q, \psi )$ orthonormal with respect to the weight zero Petersson inner product introduced above. We define

Then the Selberg eigenvalue conjecture asserts $\vartheta = 0$ , whereas Selberg only established the upper bound $\vartheta \leqslant 1/4$ . The current world record is $\vartheta \leqslant 7/64$ due to Kim–Sarnak [Reference Kim45] (see Blomer–Brumley [Reference Blomer and Brumley7] for the treatment in a more general scenario). The decomposition of the space of square-integrable, weight $\kappa $ automorphic forms on $\mathbb {H}$ with respect to the eigenspaces of the hyperbolic Laplacian involves the Eisenstein spectrum $\mathcal {E}(q, \psi )$ , which is the orthogonal complement to the space of Maaß cusp forms. It is explicitly described in terms of the Eisenstein series $E_{\mathfrak {a}}(z, 1/2+it)$ , where $\mathfrak {a}$ runs over singular cusps and $t \in \mathbb {R}$ . In this article, instead of using the classical Eisenstein series indexed by singular cusps as a basis of the continuous spectrum, we use another basis of Eisenstein series indexed by a set of parameters of the form

$$ \begin{align*} \{(\psi_{1}, \psi_{2}, f): \psi_{1} \psi_{2} = \psi, \, f \in \mathcal{B}(\psi_{1}, \psi_{2}) \}, \end{align*} $$

where $(\psi _{1}, \psi _{2})$ ranges over the pairs of characters of modulus q such that $\psi _{1} \psi _{2} = \psi $ and $\mathcal {B}(\psi _{1}, \psi _{2})$ is a certain finite set dependent on $(\psi _{1}, \psi _{2})$ . We do not need to be more explicit here, and we refer the reader to [Reference Gelbart, Jacquet, Borel and Casselman24] for an accurate definition of these parameters. The principal advantage of such a basis is that the Eisenstein series are eigenforms of the Hecke operators $T_{n}$ with $(n, q) = 1$ : we have

$$ \begin{align*} T_{n} E(z, 1/2+it, f) = \lambda_{f}(n, t) E(z, 1/2+it, f) \end{align*} $$

with

$$ \begin{align*} \lambda_{f}(n, t) = \sum_{ab = n} \psi_{1}(a) a^{it} \psi_{2}(b) b^{-it}. \end{align*} $$

For convenience, we introduce the ad hoc notation

$$ \begin{align*} i(g, z) = cz+d, \qquad j(g, z) = (cz+d)|cz+d|^{-1}, \qquad g = \begin{pmatrix} \ast & \ast \\ c & d \end{pmatrix} \in \textrm{SL}_{2}(\mathbb{R}). \end{align*} $$

We write the Fourier expansion of $f \in \mathcal {B}_{k}(q, \psi )$ around a singular cusp $\mathfrak {a}$ with a scaling matrix $\sigma _{\mathfrak {a}}$ as

$$ \begin{align*} i(\sigma_{\mathfrak{a}}, z)^{-k} f(\sigma_{\mathfrak{a}} z) = \left(\frac{(4\pi)^{k}}{\Gamma(k)} \right)^{1/2} \sum_{n = 1}^{\infty} \rho_{f \mathfrak{a}}(n) n^{(k-1)/2} e(nz). \end{align*} $$

Let $f \in \mathcal {B}_{\kappa }(q, \psi )$ be an orthonormal basis of the space of Maaß cusp forms of weight $\kappa $ with respect to $\Gamma _{0}(q)$ and central character $\psi $ . As is customary, we assume that each f is either even or odd depending on whether $T_{-1} f = f$ or $T_{-1} f = -f$ . If we denote the corresponding spectral parameter by $t_{f}$ , we have

$$ \begin{align*} j(\sigma_{\mathfrak{a}}, z)^{-\kappa} f(\sigma_{\mathfrak{a}} z) = \sqrt{\cosh(\pi t_{f})} \sum_{n \ne 0} \frac{\rho_{f \mathfrak{a}}(n)}{|n|^{1/2}} W_{\frac{n}{|n|} \frac{\kappa}{2}, it_{f}}(4\pi|n|y) e(nx) \end{align*} $$

with $W_{\alpha , \beta }$ the standard Whittaker function. For an Eisenstein series $E(z, 1/2+it, f)$ , we write

$$ \begin{align*} j(\sigma_{\mathfrak{a}}, z)^{-\kappa} E(\sigma_{\mathfrak{a}} z, 1/2+it, f) &= c_{1, f, \mathfrak{a}}(t) y^{1/2+it}+c_{2, f, \mathfrak{a}}(t) y^{1/2-it}\\ &\qquad\qquad\qquad \ \ + \sqrt{\cosh(\pi t)} \sum_{n \ne 0} \frac{\rho_{f \mathfrak{a}}(n, t)}{|n|^{1/2}} W_{\frac{n}{|n|} \frac{\kappa}{2}, it}(4\pi |n|y) e(nx). \end{align*} $$

Now let $\mathcal {B}_{\kappa }^{\ast }(q, \psi )$ be an orthonormal basis consisting of Hecke–Maaß newforms of level q and central character $\psi \pmod {q_{0}}$ normalised so that $\lambda _{f}(1) = 1$ , and we define similarly $\mathcal {B}_{k}^{\ast }(q, \psi )$ . By Atkin–Lehner–Li theory [Reference Atkin and Lehner2, Reference Atkin and Li3], we have the following direct sum decomposition as in [Reference Blomer and Milićević15]:

(2.20) $$ \begin{align} \mathcal{B}_{\kappa}(q, \psi) = \bigsqcup_{q_{1} q_{2} = q} \bigsqcup_{f \in \mathcal{B}_{\kappa}^{\ast}(q_{1}, \psi)} \mathcal{S}_{q_{2}}^{\square}(f), \end{align} $$

where an element of the orthonormal basis $\mathcal {S}_{q_{2}}^{\square }(f) = \{f_{(d)}(z): d \mid q_{2} \}$ is written as a linear combination of $f(cz)$ with $c \mid d$ . A description of this linear combination is given by Iwaniec–Luo–Sarnak [Reference Iwaniec, Luo and Sarnak39] for the squarefree level and principal nebentypus. Blomer–Milićević [Reference Blomer and Milićeić14] subsequently orthonormalised the collection of Maaß forms $\{f(dz): d \mid q_{2} \}$ for a newform f and an integer $q_{2} \geqslant 1$ , which is not necessarily squarefree. Their scheme focuses on the principal nebentypus, but the generalisation to nontrivial central characters is feasible; see [Reference Humphries31]. There are the Atkin–Lehner–Li theory for the continuous spectrum and a decomposition into spaces of oldforms analogous to equation (2.20) due to Young [Reference Young70]. We make use of equation (2.20) to apply Hecke relations and the proportionality of Fourier coefficients to the Hecke eigenvalues.

In what follows, we make an abuse of notation $\rho _{f \infty }(n) = \rho _{f}(n)$ and $\rho _{f \infty }(n, t) = \rho _{f}(n, t)$ . We consider the Fourier coefficients $\rho _{f \mathfrak {a}}(n)$ in more detail, and we stick to the case of $\mathfrak {a} \sim \infty $ . Let $f \in \mathcal {B}_{\kappa }(q, \psi )$ be any Hecke eigenform, and let $\lambda _{f}(n)$ denote the corresponding eigenvalue for $T_{n}$ , namely $T_{n} f = \lambda _{f}(n) f$ . Then we often use the Hecke multiplicativity relations for $(mn, q) = 1$ :

(2.21) $$ \begin{align} \lambda_{f}(mn) = \sum_{d \mid (m, n)} \mu(d) \psi(d) \lambda_{f} \left(\frac{m}{d} \right) \lambda_{f} \left(\frac{n}{d} \right), \qquad \lambda_{f}(m) \lambda_{f}(n) = \sum_{d \mid (m, n)} \psi(d) \lambda_{f} \left(\frac{mn}{d^{2}} \right). \end{align} $$

We see some structural beauty of equation (2.21) in Section 3.5 along with the formula $\lambda _{f}(n) = \psi (n) \overline {\lambda _{f}(n)}$ for $(n, q) = 1$ . Notice that the relation in equation (2.21) is valid for all $m, n \geqslant 1$ if f is a newform. We invoke the bounds for the Hecke eigenvalues: if f belongs to $\mathcal {B}_{k}(q, \psi )$ or is an Eisenstein series, there follows that

$$ \begin{align*} |\lambda_{f}(n)| \leqslant \tau(n) \ll_{\epsilon} n^{\epsilon} \end{align*} $$

for any $\epsilon> 0$ . For $f \in \mathcal {B}_{\kappa }(q, \psi )$ , the general upper bound is available in the form

(2.22) $$ \begin{align} |\lambda_{f}(n)| \leqslant \tau(n) n^{\vartheta} \ll_{\epsilon} n^{\vartheta+\epsilon}. \end{align} $$

There is a connection between the Fourier coefficients and the Hecke eigenvalues

(2.23) $$ \begin{align} \rho_{f}(n) = \rho_{f}(1) \lambda_{f}(n), \qquad |\rho_{f}(1)|^{2} = \frac{\varphi(q)}{2q^{2}} \frac{1}{L(1, \mathrm{Ad}^{2} f)} \end{align} $$

for $(n, q) = 1$ . We refer the reader to [Reference Duke, Friedlander and Iwaniec23, (6.14)] for the former and to [Reference Blomer6, (2.10)] for the latter. Here the adjoint square L-function is defined as

(2.24) $$ \begin{align} L(s, \mathrm{Ad}^{2} f) = \zeta(2s) \sum_{n = 1}^{\infty} \frac{\lambda_{f}(n^{2})}{n^{s}}. \end{align} $$

The formula in equation (2.23) is valid for all $n \geqslant 1$ if f is a newform.

The aforementioned formulæ for Hecke operators acting on holomorphic and Maaß cusp forms apply to the Eisenstein series since $\rho _{f}(n, t)$ are proportional to the Hecke eigenvalues $\lambda _{f}(n, t)$ (compare [Reference Duke, Friedlander and Iwaniec23, (7.13)]). We also derive the relation

$$ \begin{align*} \rho_{f}(-n, t) = \frac{\Gamma(s+\kappa/2)}{\Gamma(s-\kappa/2)} \rho_{f}(n, t). \end{align*} $$

In Section 3, we are mainly interested in primitive Dirichlet characters $\psi ^{2}$ for primitive nonquadratic characters $\psi $ modulo a prime. In this case, one observes that $\kappa (\psi ^{2}) = 0$ , obtaining in particular that

$$ \begin{align*} |\rho_{f}(n, t)|^{2} = \frac{|\lambda_{f}(n, t)|^{2}}{q|L(1+2it, \overline{\psi_{1}} \psi_{2})|^{2}}. \end{align*} $$

We also need to define the twisted automorphic L-function

(2.25)

which has meromorphic continuation to the whole complex plane $\mathbb {C}$ .

2.5 A version of the Kuznetsov formula

A spectral summation formula that we deploy here is an asymmetric trace formula relating sums of Kloosterman sums to Fourier coefficients of automorphic forms in the sense that the spectral side and the geometric side have a quite different shape. This technology is able to establish that there are considerable cancellations in sums of Kloosterman sums. Moreover, it provides a separation of variables m and n in the sum $\sum _{c} S(m, n; c)/c$ , which is conducive to obtaining additional savings in summations over m and n.

For $x> 0$ , we introduce the three integral kernels

where we have borrowed the normalisation from [Reference Topaçoğullari67]. We decided to suppress $\psi $ from the notation, and we adopt this kind of abuse of notation unless otherwise specified. For $F \in \mathcal {C}_{c}^{\infty }(\mathbb {R}_{+})$ , we introduce

$$ \begin{align*} \mathscr{L}^{\Diamond} F = \int_{0}^{\infty} \mathcal{H}^{\Diamond}(\eta, \cdot) F(\eta) \frac{d\eta}{\eta} \end{align*} $$

for $\Diamond \in \{+, -, \mathrm {hol} \}$ . With the whole notation set up, we formulate the Kloosterman summation formula, which in our case reads as follows:

Theorem 2.11 ([Reference Topaçoğullari67, Theorem 3.2])

Assume $F \in C^{3}(0, \infty )$ satisfies $x^{j} F^{(j)}(x) \ll \min (x, x^{-3/2})$ for $0 \leqslant j \leqslant 3$ . Let $\mathfrak {a}, \mathfrak {b}$ be singular cusps and $\psi \pmod {q_{0}}$ be a Dirichlet character, and let $m, n \geqslant 1$ . Then we have that

Here we set

(2.26)

(2.27)

(2.28)

(2.29)

where the sum over c on the right-hand side of equation (2.26) runs over all positive real numbers for which the Kloosterman sum $S_{\mathfrak {ab}}(m, \pm n; c; \psi )$ is non-empty.

The above formulation mimics the Kuznetsov formula that Motohashi [Reference Motohashi55] utilised to show a spectral reciprocity for the smoothed fourth moment of the Riemann zeta function. The name of the Kloosterman summation formula stems from the fact that it expresses certain sums of Kloosterman sums weighted by a function F in terms of certain sums over automorphic forms weighted by transformed functions $\mathscr {L}^{\Diamond } F$ with $\Diamond \in \{+, -, \mathrm {hol} \}$ . We emphasise that there is no delta term in the Kloosterman summation formula.

3.1 Dissection argument of Atkinson

Recall equation (1.3) and Convention 1.10. Let $\mathcal {R}_{4}^{+}$ (respectively, $\mathcal {R}_{4}^{-}$ ) be the subdomain of $\mathbb {C}^{4}$ , where all four parameters have real parts larger than (respectively, less than) one. To wit, we define

For convenience, we write

for a vector $\texttt {s} = (s_{1}, s_{2}, s_{3}, s_{4})$ . In the domain $\mathcal {R}_{4}^{+}$ , we open up the Dirichlet series to recast $\mathcal {Z}_{2}$ as

(3.1) $$ \begin{align} \int_{-\infty}^{\infty} \sum_{n_{1} = 1}^{\infty} \sum_{n_{2} = 1}^{\infty} \sum_{n_{3} = 1}^{\infty} \sum_{n_{4} = 1}^{\infty} &\frac{\chi(n_{1}n_{2}) \overline{\chi}(n_{3} n_{4})}{n_{1}^{s_{1}} n_{2}^{s_{2}} n_{3}^{s_{3}} n_{4}^{s_{4}}} \left(\frac{n_{1}n_{2}}{n_{3} n_{4}} \right)^{-it} g(t) dt \notag \\[-2pt] &\qquad\qquad \quad \ = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{\chi(m) \overline{\chi}(n) \sigma_{s_{1}-s_{2}}(m) \sigma_{s_{3}-s_{4}}(n)}{m^{s_{1}} n^{s_{3}}} \hat{g} \left(\frac{1}{2\pi} \log \frac{m}{n} \right),\\[-16pt]\nonumber \end{align} $$

where $\hat {g}$ is the Fourier transformFootnote ⁴

$$ \begin{align*} \hat{g}(\xi) = \int_{-\infty}^{\infty} g(t) e(-\xi t) dt.\\[-16pt] \end{align*} $$

Shifting the contour in equation (1.3) yields that $\mathcal {Z}_{2}$ is meromorphic over the domain

$$ \begin{align*} \mathcal{B}_{4} = \{(s_{1}, s_{2}, s_{3}, s_{4}) \in \mathbb{C}^{4}: |s_{1}|, |s_{2}|, |s_{3}|, |s_{4}| < B \},\\[-16pt] \end{align*} $$

where $B = cA$ and c is a small positive constant so that B is sufficiently large. Although it is possible to make B tend to infinity, we deal with the regime $\mathcal {B}_{4}$ for technical convenience. Since we are assuming that $\chi $ is primitive, the fourth moment $\mathcal {Z}_{2}$ is holomorphic in the vicinity of the point $\texttt {s} = (1/2, 1/2, 1/2, 1/2)$ . As an initial manipulation, we apply Atkinson’s dissection, decomposing the double sum in equation (3.1) into three parts. It therefore follows that

(3.2) $$ \begin{align} \mathcal{Z}_{2} = \mathcal{D}_{4}+\mathcal{OD}_{4}^{\dagger}+\mathcal{OD}_{4}^{\ddagger},\\[-16pt]\nonumber \end{align} $$

where $\mathcal {D}_{4}$ is the diagonal contribution corresponding to the terms with $m = n$ , whereas $\mathcal {OD}_{4}^{\dagger }$ (respectively, $\mathcal {OD}_{4}^{\ddagger }$ ) is the off-diagonal contribution corresponding to the terms with $m> n$ (respectively, $m < n$ ). Note that the case of $m < n$ is symmetric to the case of $m> n$ as follows:

(3.3) $$ \begin{align} \mathcal{OD}_{4}^{\ddagger}(s_{1}, s_{2}, s_{3}, s_{4}; g; \chi) = \overline{\mathcal{OD}_{4}^{\dagger}(\overline{s_{3}}, \overline{s_{4}}, \overline{s_{1}}, \overline{s_{2}}; g; \chi)}.\\[-16pt]\nonumber \end{align} $$

Breaking the summation in this manner leads one to certain sums of Kloosterman sums and ultimately to automorphic forms. For an explanation of the reason this is not as surprising as it initially seems, we refer the reader to the discussion in [Reference Motohashi55, §4.2].

Lemma 3.1. With the notation as above, we have that

$$ \begin{align*} \mathcal{D}_{4} = \hat{g}(0) \frac{\zeta^{q}(s_{1}+s_{3}) \zeta^{q}(s_{1}+s_{4}) \zeta^{q}(s_{2}+s_{3}) \zeta^{q}(s_{2}+s_{4})}{\zeta^{q}(s_{1}+s_{2}+s_{3}+s_{4})},\\[-16pt] \end{align*} $$

where, for any L-function, the notation $L^{q}$ signifies the removal of the Euler factor at primes dividing q.

Proof. It suffices to calculate

$$ \begin{align*} \mathcal{D}_{4} = \hat{g}(0) \sum_{(m, q) = 1} \frac{\sigma_{s_{1}-s_{2}}(m) \sigma_{s_{3}-s_{4}}(m)}{m^{s_{1}+s_{3}}},\\[-16pt] \end{align*} $$

where the Dirichlet series coefficient is a multiplicative function. Since the sum is over positive integers coprime to q, one can apply Ramanujan’s identity (see [Reference Hardy and Wright28, §17.8, Theorem 305])

$$ \begin{align*} \sum_{m = 1}^{\infty} \frac{\sigma_{\alpha}(m) \sigma_{\beta}(m)}{m^{s}} = \frac{\zeta(s) \zeta(s-\alpha) \zeta(s-\beta) \zeta(s-\alpha-\beta)}{\zeta(2s-\alpha-\beta)},\\[-16pt] \end{align*} $$

omitting the Euler factors dividing q. This formula is valid as long as $\Re (s_{i})> 1/2$ .

Note that $\hat {g}(0) = \int _{-\infty }^{\infty } g(t) dt$ is the mass of the weight function $\hat {g}_{T}(0) = T \hat {g}(0)$ , where $g_{T}(x) = g(x/T)$ with $T> 0$ . Hence, if g is compactly supported on the interval $[1, 2]$ , then $\mathrm {supp}(g_{T}) \subseteq [T, 2T]$ . Since $\mathcal {D}_{4}$ has a pole of order $4$ at $\texttt {s} = (1/2, 1/2, 1/2, 1/2)$ , we should have cancellation with a similar term coming from the off-diagonal terms. This is a common feature in the study of moment problems. The continuation of the off-diagonal contribution necessitates the full machinery of spectral theory of automorphic forms associated to Hecke congruence subgroups. We handle the second term in equation (3.2) due to the symmetry in equation (3.3), and the contribution from the third term will be incorporated at the end.

3.2 Evaluation of off-diagonal terms

For notational convenience, we call

(3.4) $$ \begin{align} G(y, s) = (1+y)^{-s} \hat{g} \left(\frac{1}{2\pi} \log(1+y) \right) = \int_{(1+\delta)} \mathring{g}(\tau, s) y^{-\tau} \frac{d\tau}{2\pi i} \end{align} $$

for suitable $\delta> 0$ , where $\mathring {g}$ is the Mellin transform

(3.5) $$ \begin{align} \mathring{g}(\tau, s) = \int_{0}^{\infty} y^{\tau-1}(1+y)^{-s} \hat{g} \left(\frac{1}{2\pi} \log(1+y) \right) dy = \Gamma(\tau) \int_{-\infty}^{\infty} \frac{\Gamma(s-\tau+it)}{\Gamma(s+it)} g(t) dt, \end{align} $$

provided $\Re (s)> \Re (\tau ) > 0$ . The following lemma is useful:

Lemma 3.2. As a function of two complex variables, $\mathring {g}(\tau , s)/\Gamma (\tau )$ is entire in $\tau $ and s. In addition, $\mathring {g}(\tau , s)$ is of rapid decay in $\tau $ as long as s and $\Re (\tau )$ are bounded, say

$$ \begin{align*} \mathring{g}(\tau, s) \ll (1+|\tau|)^{-A}. \end{align*} $$

We now embark on the computation of $\mathcal {OD}_{4}^{\dagger }$ to resolve the shifted convolution problem. Pulling the Dirichlet characters out of the summations, one sees that

(3.6) $$ \begin{align} \mathcal{OD}_{4}^{\dagger} = \sum_{a, b\!\! \pmod{q} } \overline{\chi}(a) \chi(a+b) \sum_{\substack{n \equiv a \pmod{q} \\ m \equiv b \pmod{q} }} \frac{\sigma_{s_{3}-s_{4}}(n) \sigma_{s_{1}-s_{2}}(n+m)}{n^{s_{1}+s_{3}}} G \left(\frac{m}{n}, s_{1} \right). \end{align} $$

We invoke the Ramanujan expansion (also known as the Ramanujan formula) for the divisor function, which is a precise formulation of equation (1.15). For any integers $n, q$ with $(n, q) = 1$ and $\Re (\xi ) < 0$ , we have that

(3.7) $$ \begin{align} \sigma_{\xi}(n) = \zeta^{q}(1-\xi) \sum_{(\ell, q) = 1} \ell^{\xi-1} r_{\ell}(n). \end{align} $$

This follows from the formula $r_{\ell }(n) = \sum _{d \mid (\ell , n)} d \mu (\ell /d)$ and a reversal of summations. Hence, we appeal to the fact that the divisor function emerges in the Fourier expansion of the Eisenstein series for $\operatorname {\mathrm {SL}}_{2}(\mathbb {Z})$ . On account of the presence of $\chi (a+b)$ , we can assume $(n+m, q) = 1$ . Upon applying equation (3.7), the right-hand side of equation (3.6) equals

$$ \begin{align*} \zeta^{q}(1-s_{1}+s_{2}) \sum_{a, b\! \pmod{q} } \overline{\chi}(a) \chi(a+b) \sum_{(\ell, q) = 1} \ell^{s_{1}-s_{2}-1} \sum_{\substack{n \equiv a \pmod{q} \\ m \equiv b \pmod{q} }} \frac{r_{\ell}(n+m) \sigma_{s_{3}-s_{4}}(n)}{n^{s_{1}+s_{3}}} G \left(\frac{m}{n}, s_{1} \right). \end{align*} $$

In order to simplify the innermost sums over m and n, we detect the congruences modulo q in an additive manner, obtaining

$$ \begin{align*} &\mathcal{OD}_{4}^{\dagger} = \zeta^{q}(1-s_{1}+s_{2}) q^{-2} \sum_{a, b\! \pmod{q} } \overline{\chi}(a) \chi(a+b) \sum_{(\ell, q) = 1} \ell^{s_{1}-s_{2}-1} \\ \times &\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sideset{}{^{\ast}} \sum_{h\! \pmod{\ell}} \sum_{r \mid q} \sideset{}{^{\ast}} \sum_{c\! \pmod{r}} \!\sum_{d\! \pmod{q} }\! e_{r} \left(c(n-a) \right) e_{q} \left(d(m-b) \right) e_{\ell}(h(n+m)) \frac{\sigma_{s_{3}-s_{4}}(n)}{n^{s_{1}+s_{3}}} G \left(\frac{m}{n}, s_{1} \right). \end{align*} $$

The reason for the occurrence of the sum over r is to make subsequent formulæ cleaner. If we execute the change of variables $(a, b) \mapsto (-a, -b)$ , then the sum does not change according to whether $\chi $ is even or odd, and we use equation (3.4) to show that

(3.8) $$ \begin{align} \mathcal{OD}_{4}^{\dagger} &= \zeta^{q}(1-s_{1}+s_{2}) q^{-2} \nonumber\\&\quad \times \sum_{a, b\! \pmod{q} } \sum_{r \mid q} \ \sideset{}{^{\ast}} \sum_{c\! \pmod{r}} \sum_{d\! \pmod{q} } \overline{\chi}(a) \chi(a+b) e_{r}(ac) e_{q}(bd) \sum_{(\ell, q) = 1} \ell^{s_{1}-s_{2}-1} \notag\\ &\quad\times \sideset{}{^{\ast}} \sum_{h\! \pmod{\ell}} \int_{(1+\delta)} \mathring{g}(\tau, s_{1}) \zeta^{(h/\ell+d/q)}(\tau) D_{2} \left(s_{1}+s_{3}-\tau, s_{3}-s_{4}; \frac{h}{\ell}+\frac{c}{r} \right) \frac{d\tau}{2\pi i}, \end{align} $$

where the right-hand side of equation (3.8) is absolutely convergent in a suitable domain such as

$$ \begin{align*} \mathcal{R}_{4, \delta} = \{(s_{1}, s_{2}, s_{3}, s_{4}) \in \mathcal{R}_{4}^{+}: \Re(s_{1}+s_{3})> 2+2\delta, \, \Re(s_{1})+1 < \Re(s_{2}), \, |s_{3}-s_{4}| < \delta \} \end{align*} $$

for an arbitrary but fixed constant $\delta $ . We then want to shift the contour to the right. In anticipation of the future application of the functional equation of the Estermann zeta function (Theorem 2.6), we define

(3.9) $$ \begin{align} \mathcal{E}_{4} = \left\{(s_{1}, s_{2}, s_{3}, s_{4}) \in \mathcal{B}_{4}: \Re(s_{1}+s_{3}) < \frac{B}{3}, \ \Re(s_{1}+s_{4}) < \frac{B}{3}, \ \Re(s_{1}+s_{2}+s_{3}+s_{4})> 3B \right\}. \end{align} $$

We ascertain that $\mathcal {R}_{4, \delta } \cap \mathcal {E}_{4} \ne \emptyset $ if $\delta $ is small and confine the variables $(s_{1}, s_{2}, s_{3}, s_{4})$ to be in the codomain $\mathcal {E}_{4}$ , and then our shift of the contour in equation (3.8) to $\Re (\tau ) = B$ results in

(3.10) $$ \begin{align} \sideset{}{^{\ast}} \sum_{h\! \pmod{\ell}} &\bigg\{\int_{(1+\delta)}-\int_{(B)} \bigg\} \mathring{g}(\tau, s_{1}) \zeta^{(h/\ell+d/q)}(\tau) D_{2} \left(s_{1}+s_{3}-\tau, s_{3}-s_{4}; \frac{h}{\ell}+\frac{c}{r} \right) \frac{d\tau}{2\pi i} \notag \\ &\qquad \ = (\ell r)^{s_{3}-s_{4}-1} \zeta(1-s_{3}+s_{4}) \mathring{g}(s_{1}+s_{3}-1, s_{1}) \sum_{n = 1}^{\infty} r_{\ell}(n) e_{q}(nd) n^{1-s_{1}-s_{3}} \notag \\ &\qquad\qquad\quad \ + (\ell r)^{s_{4}-s_{3}-1} \zeta(1+s_{3}-s_{4}) \mathring{g}(s_{1}+s_{4}-1, s_{1}) \sum_{n = 1}^{\infty} r_{\ell}(n) e_{q}(nd) n^{1-s_{1}-s_{4}}. \end{align} $$

Since the above polar terms do not contain the parameter c, one may express the sum over c in equation (3.8) as $r_{r}(a)$ , which leads one to a further simplification of them. At this stage, we notice that

(3.11) $$ \begin{align} \sum_{a, b\! \pmod{q} } \overline{\chi}(a) \chi(a+b) e_{q}(ac+bd) = \tau(\chi, d) \overline{\tau(\chi, d-c)} = \overline{\chi}(d) \chi(d-c) q \end{align} $$

when $r = q$ . This evaluation will not be used as the expanded sums are more amenable. Consequently, after some rearrangements, our formula is transformed into

$$ \begin{align*} \mathcal{OD}_{4}^{\dagger} & = (\text{Polar Terms})+q^{-2} \sum_{a, b, d \pmod{q}} \sum_{r \mid q} \zeta^{r}(1-s_{1}+s_{2}) \ \sideset{}{^{\ast}} \sum_{c\! \pmod{r}} \overline{\chi}(a) \chi(a+b) e_{r}(ac) e_{q}(bd)\\ &\quad \ \times \sum_{(\ell, r) = 1} \ell^{s_{1}-s_{2}-1} \sideset{}{^{\ast}} \sum_{h\! \pmod{\ell}} \int_{(B)} \mathring{g}(\tau, s_{1}) \zeta^{(h/\ell+d/q)}(\tau) D_{2} \left(s_{1}+s_{3}-\tau, s_{3}-s_{4}; \frac{h}{\ell}+\frac{c}{r} \right) \frac{d\tau}{2\pi i}, \end{align*} $$

which is now in shape to make use of Poisson summation twice, namely Voronoĭ summation once. This is because we have $\Re (s_{1}+s_{3}-\tau ) < 0$ and $\Re (s_{1}+s_{4}-\tau ) < 0$ ; so we replace the Estermann zeta function $D_{2}(s_{1}+s_{3}-\tau , s_{3}-s_{4}; h/\ell +c/r)$ with the absolutely convergent series exhibited by equation (2.14).

3.3 Utilisation of Voronoĭ summation

This subsection aims at applying the functional equation to the sum over m and simplifying the resulting expression to a certain sum of Kloosterman sums of the form $S_{\mathfrak {ab}}(m, \pm n; c; \psi )$ . To this end, the following elementary formula is necessary: if we set $v = hr+c \ell $ with $(v, \ell r) = 1$ , then

(3.12) $$ \begin{align} \overline{v} = \overline{hr+c \ell} \equiv \overline{h} r \overline{r}^{2}+\overline{c} \ell \overline{\ell}^{2} \pmod{\ell r} \end{align} $$

with $h \overline {h} \equiv 1$ , $r \overline {r} \equiv 1 \pmod {\ell }$ and $c \overline {c} \equiv 1$ , $\ell \overline {\ell } \equiv 1 \pmod {r}$ . This holds under the assumption $(\ell , r) = 1$ . The application of Theorem 2.6 hence leads one to

$$ \begin{align*} \mathcal{OD}_{4}^{\dagger} &= (\text{Polar Terms})+2q^{-2} \sum_{a, b, d \pmod{q} } \sum_{r \mid q} \frac{\zeta^{r}(1-s_{1}+s_{2})}{r} \ \sideset{}{^{\ast}} \sum_{c\! \pmod{r}} \overline{\chi}(a) \chi(a+b) e_{r}(ac) e_{q}(bd)\\ & \qquad \times \sum_{(\ell, r) = 1} \ell^{s_{1}-s_{2}-2} \sideset{}{^{\ast}} \sum_{h\! \pmod{\ell}} \int_{(B)} \left(\frac{2\pi}{\ell r} \right)^{2s_{1}+s_{3}+s_{4}-2\tau-2} \zeta^{(h/\ell+d/q)}(\tau) \Gamma(1+\tau-s_{1}-s_{3})\\ & \qquad \times \Gamma(1+\tau-s_{1}-s_{4}) \bigg\{D_{2} \left(1+\tau-s_{1}-s_{3}, s_{4}-s_{3}; \frac{\overline{h} \overline{r}^{2}}{\ell} + \frac{\overline{c} \overline{\ell}^{2}}{r} \right) \cos \left(\frac{\pi(s_{3}-s_{4})}{2} \right)\\ & \qquad - D_{2} \left(1+\tau-s_{1}-s_{3}, s_{4}-s_{3}; - \frac{\overline{h} \overline{r}^{2}}{\ell}-\frac{\overline{c} \overline{\ell}^{2}}{r} \right) \cos \left(\pi \left(\frac{s_{3}+s_{4}}{2}+s_{1}-\tau \right) \right) \bigg\} \mathring{g}(\tau, s_{1}) \frac{d\tau}{2\pi i}. \end{align*} $$

We expand the integrand as the Dirichlet series once again so that the sums over c and h boil down to two Kloosterman sums, whence we are left with

(3.13) $$ \begin{align} \begin{split} \mathcal{OD}_{4}^{\dagger} &= (\text{Polar Terms})+2 (2\pi)^{s_{1}-s_{2}-1} q^{-2} \sum_{a, b, d \pmod{q} } \overline{\chi}(a) \chi(a+b) e_{q}(bd)\\ & \qquad \times \sum_{r \mid q} r^{s_{2}-s_{1}} \zeta^{r}(1-s_{1}+s_{2}) \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} m^{(1-s_{1}-s_{2}-s_{3}-s_{4})/2} n^{(s_{1}-s_{2}+s_{3}-s_{4}-1)/2}\\ & \qquad \times e_{q}(md) \sigma_{s_{4}-s_{3}}(n) \sum_{\pm} \sum_{(\ell, r) = 1} \frac{1}{\ell} S(m, \pm n \overline{r}^{2}; \ell) S(a, \pm n \overline{\ell}^{2}; r) \Psi_{\texttt{s}}^{\pm} \left(\frac{4\pi \sqrt{mn}}{\ell r} \right), \end{split} \end{align} $$

where

(3.14)

(3.15)

The expression in equation (3.13) affirmatively answers Motohashi’s conjecture spelled out in his article [Reference Motohashi, Greaves, Harman and Huxley54] on the reciprocity formula for the second moment of Dedekind zeta functions. One ascertains that the integrand in equation (3.14) has rapid decay in $\tau $ , whereas the definition in equation (3.15) requires the properties shown in Lemma 3.2. Convention 1.10 therefore applies to control $\Psi _{\texttt {s}}^{-}(x)$ , which involves the opposite sign case of Kloosterman sums. This dominates the bulk of the evaluation of $\mathcal {OD}_{4}^{\dagger }$ . We ensure this phenomenon in Proposition 3.5.

We then manage to simplify the product of the Kloosterman sums appearing in equation (3.13). It is desirable to collapse the second Kloosterman sum to perform the separation of variables. To this end, we write $n = n_{0} n^{\prime }$ with $n_{0} \mid r^{\infty }$ and $(n^{\prime }, r) = 1$ . We are in a position to use Lemma 2.3, getting

(3.16) $$ \begin{align} S(a, \pm n \overline{\ell}^{2}; r) = \frac{1}{\varphi(r)} \sum_{\psi \pmod{r}} \psi(\ell)^{2} \overline{\psi}(\pm an^{\prime}) \tau(\psi) \tau(\psi, n_{0}), \end{align} $$

where the condition $(a, r) = 1$ was used in view of the appearance of the character $\overline {\chi }(a)$ and $\tau (\psi )$ is the Gauß sum associated to $\psi $ . Dropping the primes for notational simplicity, we observe that $\mathcal {OD}_{4}^{\dagger }$ equals the polar terms plus (note that the condition $(n, r) = 1$ is encoded by $\overline {\psi }(n)$ )

(3.17) $$ \begin{align} &2 (2\pi)^{s_{1}-s_{2}-1} q^{-2} \sum_{a, b, d \pmod{q} } \overline{\chi}(a) \chi(a+b) e_{q}(bd) \sum_{r \mid q} \frac{r^{1-s_{1}+s_{2}}}{\varphi(r)} \zeta^{r}(1-s_{1}+s_{2}) \sum_{\pm} \sum_{\psi \pmod{r}} \notag\\ \times &\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sum_{n_{0} \mid r^{\infty}} \frac{\overline{\psi}(\pm an) \tau(\psi) \tau(\psi, n_{0}) e_{q}(md) \sigma_{s_{4}-s_{3}}(n_{0} n)} {m^{(s_{1}+s_{2}+s_{3}+s_{4}-1)/2} (n_{0} n)^{(1-s_{1}+s_{2}-s_{3}+s_{4})/2}} \sum_{(\ell, r) = 1} \psi(\ell)^{2} \frac{S(m, \pm n_{0} n \overline{r}^{2}; \ell)}{\ell r} \Psi_{\texttt{s}}^{\pm} \left(\frac{4\pi \sqrt{mn_{0} n}}{\ell r} \right), \end{align} $$

In order to adapt the Kloosterman summation formula in its suitable form, there are two roads to proceed. The first route is to represent $S(m, \pm n_{0} n \overline {r}^{2}; \ell )$ by means of the Kloosterman sum associated to the $(\infty , 0)$ cusp-pair and apply Theorem 2.11 to the sum over $\ell $ . The second is to extract the principal and quadratic character from the $\psi $ -sum in equation (3.16) and then replace $S(m, \pm n_{0} n \overline {r}^{2}; \ell )$ with the twisted Kloosterman sum; namely, we make use of the formula due to Blomer–Milićević [Reference Blomer and Milićević15, black(2.2)]:

(3.18) $$ \begin{align} \sum_{(\ell, r) = 1} \psi(\ell)^{2} \frac{S(m, \pm n_{0} n \overline{r}^{2}; \ell)}{\ell r} \Psi_{\texttt{s}}^{\pm} \left(\frac{4\pi \sqrt{mn_{0} n}}{\ell r} \right) = \frac{\psi(m)^{2}}{\tau(\psi^{2})} \sum_{d \mid r} \mu(d) \sum_{dr \mid c} \frac{S_{\psi^{2}}(m, \pm n_{0} n; c)}{c} \Psi_{\texttt{s}}^{\pm} \left(\frac{4\pi \sqrt{mn_{0} n}}{c} \right), \end{align} $$

where $S_{\psi }(m, n; c)$ is defined in equation (2.17) and $\psi $ denotes a primitive nonquadratic character modulo r so that $\psi ^{2}$ is primitive. This is where the assumption of q being prime is used, since the square of a nonquadratic character is not necessarily primitive (see Remark 3.4). Nevertheless, one may remove this assumption via the use of an extended form of equation (3.18). Note that equation (3.18) is only available when $(m, r) = 1$ . As shall be seen later, the Dirichlet character $\overline {\psi }(m)$ arises if we calculate the character sums in equation (3.17). We need to take the second route for technical simplicity. The off-diagonal term $\mathcal {OD}_{4}^{\dagger }$ is thus equal to the polar terms plus

$$ \begin{align*} &2 (2\pi)^{s_{1}-s_{2}-1} q^{-2} \sum_{a, b, c\! \pmod{q} } \overline{\chi}(a) \chi(a+b) e_{q}(bc) \sum_{r \mid q} \frac{r^{1-s_{1}+s_{2}}}{\varphi(r)} \zeta^{r}(1-s_{1}+s_{2})\\ &\qquad\quad \times \sum_{\pm} \Bigg\{\ \sideset{}{^{\#}} \sum_{\psi \pmod{r}} \frac{\overline{\psi}(\pm a) \tau(\psi)}{\tau(\psi^{2})} \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sum_{n_{0} \mid r^{\infty}} \frac{\psi(m)^{2} \overline{\psi}(n) \tau(\psi, n_{0}) e_{q}(mc) \sigma_{s_{4}-s_{3}}(n_{0} n)} {m^{(s_{1}+s_{2}+s_{3}+s_{4}-1)/2} (n_{0} n)^{(1-s_{1}+s_{2}-s_{3}+s_{4})/2}}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \times \sum_{d \mid r} \mu(d) \sum_{dr \mid c} \frac{S_{\psi^{2}}(m, \pm n_{0} n; c)}{c} \Psi_{\texttt{s}}^{\pm} \left(\frac{4\pi \sqrt{mn_{0} n}}{c} \right)+\mathcal{R} \Bigg\}, \end{align*} $$

where $\#$ on the $\psi $ -sum means that the sum ranges over all primitive nonquadratic characters modulo r and $\mathcal {R}$ serves as the remainder term

$$ \begin{align*} \mathcal{R} = \sideset{}{^{\flat}} \sum_{\psi \pmod{r}} \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sum_{n_{0} \mid r^{\infty}} &\frac{\psi(\pm na) \tau(\psi) \tau(\psi, n_{0}) e_{q}(mc) \sigma_{s_{4}-s_{3}}(n_{0} n)} {m^{(s_{1}+s_{2}+s_{3}+s_{4}-1)/2} (n_{0} n)^{(1-s_{1}+s_{2}-s_{3}+s_{4})/2}}\\ &\qquad\qquad\qquad\qquad\qquad\quad \ \times \sum_{(\ell, r) = 1} \frac{S(m, \pm n_{0} n \overline{r}^{2}; \ell)}{\ell r} \Psi_{\texttt{s}}^{\pm} \left(\frac{4\pi \sqrt{mn_{0} n}}{\ell r} \right). \end{align*} $$

Here $\flat $ on the $\psi $ -sum means that the sum ranges over the principal and the quadratic character modulo r. Since it follows that $\tau (\psi , n_{0}) = \overline {\psi }(n_{0}) \tau (\psi )$ in the case where $\psi $ is a primitive nonquadratic character, we can take $n_{0} = 1$ due to the condition $n_{0} \mid r^{\infty }$ . In order to reduce the character sums over $a, b, c$ , we start with simplifying the c-sum, obtaining

$$ \begin{align*} \sum_{a, b, c\! \pmod{q} } \overline{\chi}(a) \overline{\psi}(a) \chi(a+b) e_{q}((b+m)c) = \psi(-1) q \sum_{a\! \pmod{q} } \overline{\chi}(a) \overline{\psi}(a) \chi(a+m). \end{align*} $$

When $\chi \psi $ is primitive, one can set $\psi _{1} = \overline {\chi \psi }$ and $\psi _{2} = \chi $ in Lemma 2.2. To be more general, we deduce

$$ \begin{align*} \sum_{a\! \pmod{q} } \overline{\chi}(a) \overline{\psi}(a) \chi(a+m) = \frac{1}{\tau(\overline{\chi})} \sum_{a, b\! \pmod{q} } \overline{\chi}(ab) \overline{\psi}(a) e \left(\frac{(a+m)b}{q} \right) = \frac{\tau(\overline{\chi \psi}) \tau(\psi, m)}{\tau(\overline{\chi})}. \end{align*} $$

Since the sum over primitive nonquadratic characters is empty when $r = 1$ , we have proven the following:

Proposition 3.3. For any primitive Dirichlet character $\chi $ modulo a prime q, the function $\mathcal {OD}_{4}^{\dagger }$ can be meromorphically continued to the domain $\mathcal {E}_{4}$ , and there we have

$$ \begin{align*} \mathcal{OD}_{4}^{\dagger} = \mathcal{P}+\sum_{\pm} \left\{\mathcal{J}_{\pm}+\mathcal{E}_{\pm} \right\}. \end{align*} $$

Here for the multiplicative function

we set

$$ \begin{align*} \mathcal{P} &= A_{q}(s_{1}, s_{2}, s_{3}, s_{4}) \mathring{g}(s_{1}+s_{3}-1, s_{1}) \frac{\zeta^{q}(1-s_{1}+s_{2}) \zeta^{q}(1-s_{3}+s_{4}) \zeta(s_{2}+s_{4}) \zeta(s_{1}+s_{3}-1)}{\zeta^{q}(s_{2}+s_{4}-s_{1}-s_{3}+2)}\\ &\quad + A_{q}(s_{1}, s_{2}, s_{4}, s_{3}) \mathring{g}(s_{1}+s_{4}-1, s_{1}) \frac{\zeta^{q}(1-s_{1}+s_{2}) \zeta^{q}(1+s_{3}-s_{4}) \zeta(s_{2}+s_{3}) \zeta(s_{1}+s_{4}-1)}{\zeta^{q}(s_{2}+s_{3}-s_{1}-s_{4}+2)}, \end{align*} $$

(3.19) $$ \begin{align} \mathcal{J}_{\pm} &= \frac{2\zeta^{q}(1-s_{1}+s_{2})}{\tau(\overline{\chi}) \varphi(q) q} \left(\frac{2\pi}{q} \right)^{s_{1}-s_{2}-1} \ \sideset{}{^{\#}} \sum_{\psi \pmod{q} } \psi(\mp 1) \tau(\psi) \tau(\overline{\chi \psi}) J(\psi, \psi) \sum_{d \mid q} \mu(d) \notag\\ &\quad \times \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \psi(m) \overline{\psi}(n) m^{(1-s_{1}-s_{2}-s_{3}-s_{4})/2} n^{(s_{1}-s_{2}+s_{3}-s_{4}-1)/2} \sigma_{s_{4}-s_{3}}(n) \, \mathcal{O}_{\infty \infty}^{dq}(m, \pm n; \Psi_{\texttt{s}}^{\pm}; \psi^{2}), \end{align} $$

(3.20) $$ \begin{align} &\mathcal{E}_{\pm} = \frac{2(2\pi)^{s_{1}-s_{2}-1}}{\tau(\overline{\chi}) q} \sum_{r \mid q} \frac{r^{1-s_{1}+s_{2}}}{\varphi(r)} \zeta^{r}(1-s_{1}+s_{2}) \ \sideset{}{^{\flat}} \sum_{\psi \pmod{r}} \psi(\mp 1) \tau(\psi) \tau(\overline{\chi} \psi) \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sum_{n_{0} \mid r^{\infty}} \psi(n) \notag\\ \times\, &\tau(\psi, m) \tau(\psi, n_{0}) m^{(1-s_{1}-s_{2}-s_{3}-s_{4})/2} (n_{0} n)^{(s_{1}-s_{2}+s_{3}-s_{4}-1)/2} \sigma_{s_{4}-s_{3}}(n_{0} n) \, \mathcal{O}_{\infty 0}^{r^{2}}(m, \pm n_{0} n; \Psi_{\texttt{s}}^{\pm}; \psi_{0}), \end{align} $$

where $\mathcal {O}_{\mathfrak {ab}}^{q}(m, \pm n; \Psi _{\texttt {s}}^{\pm }; \psi )$ is defined in Theorem 2.11.

Remark 3.4. The square of a primitive nonquadratic Dirichlet character of composite modulus having at least two prime factors is not primitive. For instance, one may choose a primitive quadratic character $\psi _{1}$ (respectively, nonquadratic character $\psi _{2}$ ) modulo $3$ (respectively, modulo $5$ ) and combine them to obtain a character $\psi _{1} \psi _{2}$ modulo $15$ as follows:

The multiplication of Dirichlet characters of different moduli gives a Dirichlet character modulo a least common multiple. We define $\psi = \psi _{1} \psi _{2}$ , which is nonquadratic since it involves i and $-i$ as values and is primitive since it cannot be induced from a Dirichlet character modulo $1$ , $3$ , or $5$ . However, $\psi ^{2} = \psi _{1}^{2} \psi _{2}^{2}$ is induced from a quadratic character modulo 5. This is why we assume that q is prime in this work.

Proof. For the first assertion, one can check that both series in equations (3.19) and (3.20) converge absolutely and uniformly in $\mathcal {E}_{4}$ , proving the meromorphic continuation of $\mathcal {OD}_{4}^{\dagger }$ to $\mathbb {C}^{4}$ . A remarkable point is that we only need the Weil bound in equation (2.5) for Kloosterman sums. For the second assertion, it suffices to compute the polar terms. We decompose $\mathcal {P} = \mathcal {P}_{1}+\mathcal {P}_{2}$ , where $\mathcal {P}_{1}$ (respectively, $\mathcal {P}_{2}$ ) stems from the first term (respectively, second term) on the right-hand side of equation (3.10). First of all, we remark that the polar contribution $\mathcal {P}$ is expressed as

$$ \begin{align*} \zeta^{q}(1-s_{1}+s_{2}) q^{-2} &\sum_{a, b\! \pmod{q} } \sum_{r \mid q} \ \sideset{}{^{\ast}} \sum_{c\! \pmod{r}} \sum_{d\! \pmod{q} } \overline{\chi}(a) \chi(a+b)\\ &\qquad\qquad\qquad \times e_{r}(ac) e_{q}(bd) \sum_{(\ell, q) = 1} \ell^{s_{1}-s_{2}-1} \times (\text{RHS of equation (3.10)}). \end{align*} $$

In the following lines, we deal with $\mathcal {P}_{1}$ . The sum over $\ell $ is reduced to

$$ \begin{align*} \sum_{(\ell, q) = 1} \ell^{s_{1}-s_{2}+s_{3}-s_{4}-2} r_{\ell}(n) = \zeta^{q}(s_{2}+s_{4}-s_{1}-s_{3}+2)^{-1} \sigma_{s_{1}-s_{2}+s_{3}-s_{4}-1}(n). \end{align*} $$

We are in a position to manipulate the exponential sums. The sum over d gives rise to

$$ \begin{align*} \sum_{d\! \pmod{q} } e_{q}((b+n)d) = \begin{cases} q & \text{if } b \equiv -n \pmod{q} ,\\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

This renders

$$ \begin{align*} \sum_{a\! \pmod{q} } \overline{\chi}(a) \chi(a-n) r_{r}(a) = \frac{\mu(r)}{\tau(\overline{\chi})} \sum_{a, b\! \pmod{q} } \overline{\chi}(ab) e_{q}((a+n)b) = \mu(r) r_{q}(n). \end{align*} $$

In this manner, the sums over $a, b, c, d$ can be eliminated. In general, the sum over r turns into

$$ \begin{align*} \sum_{r \mid q} \mu(r) r^{s_{3}-s_{4}-1} = \prod_{p \mid q} \left(1-\frac{1}{p^{1-s_{3}+s_{4}}} \right), \end{align*} $$

which yields $\zeta ^{q}(1-s_{3}+s_{4})$ when combined with $\zeta (1-s_{3}+s_{4})$ . We hence obtain

$$ \begin{align*} \mathcal{P}_{1} = \frac{\mathring{g}(s_{1}+s_{3}-1, s_{1})}{q} \frac{\zeta^{q}(1-s_{1}+s_{2}) \zeta^{q}(1-s_{3}+s_{4})}{\zeta^{q}(s_{2}+s_{4}-s_{1}-s_{3}+2)} \sum_{n = 1}^{\infty} \frac{\sigma_{s_{1}-s_{2}+s_{3}-s_{4}-1}(n) r_{q}(n)}{n^{s_{1}+s_{3}-1}}. \end{align*} $$

Having mentioned this expression, we need to calculate the the sum over n via Lemma 2.5, getting

$$ \begin{align*} \sum_{n = 1}^{\infty} \frac{\sigma_{w}(n)}{n^{s}} \sum_{d \mid (n, q)} \mu \left(\frac{q}{d} \right) d & = \sum_{d \mid q} \mu \left(\frac{q}{d} \right) d^{1-s} \sum_{n = 1}^{\infty} \frac{1}{n^{s}} \sum_{c \mid (n, d)} \mu(c) \sigma_{w} \left(\frac{n}{c} \right) \sigma_{w} \left(\frac{d}{c} \right) c^{w}\\ & = \sum_{d \mid q} \mu \left(\frac{q}{d} \right) d^{1-s} \sum_{c \mid d} \mu(c) c^{w-s} {\sigma_{w} \left(\frac{d}{c} \right)} \sum_{n = 1}^{\infty} \frac{\sigma_{w}(n)}{n^{s}}\\ & = \zeta(s) \zeta(s-w) \sum_{c \mid q} \mu(c) c^{1+w-2s} \sum_{d \mid q/c} \mu \left(\frac{q}{cd} \right) d^{1-s} \sigma_{w}(d), \end{align*} $$

where $s = s_{1}+s_{3}-1$ and $w = s_{1}-s_{2}+s_{3}-s_{4}-1$ . Notice that the right-hand side can be written as a Dirichlet convolution of multiplicative functions. Recall that the Dirichlet convolution is defined as

$$ \begin{align*} (f \ast g)(m) = \sum_{d \mid m} f(d) g \left(\frac{m}{d} \right), \end{align*} $$

which is an associative and commutative binary operation on arithmetic functions. If f and g are assumed to be multiplicative, so is $f \ast g$ . Setting and , we have that

$$ \begin{align*} \sum_{n = 1}^{\infty} \frac{\sigma_{w}(n) r_{q}(n)}{n^{s}} = (f \ast g \ast \mu)(q) \zeta(s) \zeta(s-w). \end{align*} $$

This finishes the proof of Proposition 3.3.

3.4 Utilisation of the Kuznetsov formula

We now apply the automorphic machinery in Section 2 to our sums of Kloosterman sums along with a careful analysis of integrals involving various Bessel functions. We focus on the treatment of $\mathcal {O}_{\infty \mathfrak {b}}^{dq}(m, \pm n; \Psi _{\texttt {s}}^{\pm }; \psi )$ with $\mathfrak {b} = \infty , 0$ in an across-the-board manner and implement the substitutions at the end to calculate $\mathcal {J}_{\pm }$ and $\mathcal {E}_{\pm }$ . The three-time differentiability and the decay condition at $x = 0$ in Theorem 2.11 are fulfilled. Moreover, the decay condition at infinity follows by shifting the contours $\Re (s) = B$ in equations (3.14) and (3.15) to the right and then applying Lemma 3.2. We forthwith derive the spectral decomposition

(3.21)

Complying with Motohashi’s book [Reference Motohashi55], one observes how the case of the plus sign contributes. In order to reduce the integral transform $\mathscr {L}^{+}$ , we consider, in the light of the definition in equation (3.14), the double integral

(3.22) $$ \begin{align} \int_{0}^{\infty} J_{2it}(\eta) \int_{(B)} \Gamma(1+\tau-s_{1}-s_{3}) \Gamma(1+\tau-s_{1}-s_{4}) \left(\frac{\eta}{2} \right)^{s_{1}+s_{2}+s_{3}+s_{4}-2-2\tau} \mathring{g}(\tau, s_{1}) d\tau d\eta, \end{align} $$

where $\mathring {g}$ is the Mellin transform in equation (3.5). This double integral is holomorphic in the domain

(3.23) $$ \begin{align} \left\{(s_{1}, s_{2}, s_{3}, s_{4}) \in \mathbb{C}^{4}: \begin{array}{l} \Re(s_{1}+s_{3}) < 1+B, \, \Re(s_{1}+s_{4}) < 1+B,\\ 1+2B < \Re(s_{1}+s_{2}+s_{3}+s_{4}) < 3/2+2A \end{array} \right\}, \end{align} $$

which contains $\mathcal {E}_{4}$ defined in equation (3.9) and $A> B$ is the same as in Convention 1.10. To prove this fact, we divide equation (3.22) into two parts according as $0 \leqslant \eta < 1$ and $\eta \geqslant 1$ and note that

$$ \begin{align*} J_{2it}(\eta) \ll \begin{cases} 1 & \text{as } \eta \to 0,\\ \eta^{-1/2} & \text{as } \eta \to \infty, \end{cases} \end{align*} $$

where we implicitly assumed that t is real. The first part is clearly holomorphic in the domain of equation (3.23). To estimate the second part, we move the contour in the $\tau $ -integral to $\Re (\tau ) = A$ . We would like to consider the subdomain of equation (3.23) where we have $1+2B < \Re (s_{1}+s_{2}+s_{3}+s_{4}) < 3/2+2B$ . There the integral in equation (3.22) is absolutely and uniformly convergent in view of the power series expansion of the J-Bessel function

$$ \begin{align*} J_{\nu}(y) &= \sqrt{\frac{2}{\pi y}} \bigg\{\cos \left(y-\frac{\pi}{2} \left(\nu+\frac{1}{2} \right) \right) \sum_{m = 0}^{M} (-1)^{m} \begin{pmatrix} \nu-1/2 \\ 2m \end{pmatrix} \frac{\Gamma(\nu+1/2+2m)}{\Gamma(\nu+1/2)(2y)^{2m}}\\ & \quad - \sin \left(y-\frac{\pi}{2} \left(\nu+\frac{1}{2} \right) \right) \sum_{m = 0}^{M} (-1)^{m} \begin{pmatrix} \nu-1/2 \\ 2m+1 \end{pmatrix} \frac{\Gamma(\nu+3/2+2m)}{\Gamma(\nu+1/2)(2y)^{2m+1}} \bigg\}+\mathrm{O}(y^{-\Re(\nu)-3/2-2M}), \end{align*} $$

provided $\Re (\nu )> -2M-3/2$ . Hence, the double integral in equation (3.22) equals

(3.24) $$ \begin{align} \int_{(B)} \frac{\Gamma \left(\frac{s_{1}+s_{2}+s_{3}+s_{4}-1}{2}-\tau+it \right)} {\Gamma \left(\frac{3-s_{1}-s_{2}-s_{3}-s_{4}}{2}+\tau+it \right)} \Gamma(1+\tau-s_{1}-s_{3}) \Gamma(1+\tau-s_{1}-s_{4}) \mathring{g}(\tau, s_{1}) d\tau, \end{align} $$

where we have used the formula

$$ \begin{align*} \int_{0}^{\infty} J_{\nu}(\eta) \left(\frac{\eta}{2} \right)^{-\mu} d\eta = \Gamma \left(\frac{1+\nu-\mu}{2} \right) \Gamma \left(\frac{1+\nu+\mu}{2} \right)^{-1} \end{align*} $$

valid for $1/2 < \Re (\mu ) < 1+\Re (\nu )$ . Since the integral in equation (3.24) is regular in equation (3.23), the analytic continuation implies that the double integral in equation (3.22) is equal to equation (3.24) throughout the domain of equation (3.23) or $\mathcal {E}_{4}$ . At this stage, we note that the following identities hold:

$$ \begin{align*} \frac{\Gamma(a-\tau+it)}{\Gamma(1-a+\tau+it)}-\frac{\Gamma(a-\tau-it)}{\Gamma(1-a+\tau-it)} & = \frac{2}{\pi i} \sinh(\pi t) \cos(\pi(a-\tau)) \Gamma(a-\tau+it) \Gamma(a-\tau-it),\\ \frac{\Gamma(a-\tau+it)}{\Gamma(1-a+\tau+it)}+\frac{\Gamma(a-\tau-it)}{\Gamma(1-a+\tau-it)} & = \frac{2}{\pi} \cosh(\pi t) \sin(\pi(a-\tau)) \Gamma(a-\tau+it) \Gamma(a-\tau-it). \end{align*} $$

which follows from the functional equation $\Gamma (s) \Gamma (1-s) = \pi /\sin (\pi s)$ . Substituting $a = (s_{1}+s_{2}+s_{3}+s_{4}-1)/2$ , we obtain after some rearrangement thatFootnote ⁵

$$ \begin{align*} \mathscr{L}^{+} \Psi_{\texttt{s}}^{+}(t) &= (it \coth(\pi t))^{\kappa} \cos \left(\frac{\pi(s_{3}-s_{4})}{2} \right) \int_{(B)} \sin \left(\frac{\pi(s_{1}+s_{2}+s_{3}+s_{4}-\kappa-2\tau)}{2} \right)\\ & \qquad \times \Gamma \left(\frac{s_{1}+s_{2}+s_{3}+s_{4}-1}{2}+it-\tau \right) \Gamma \left(\frac{s_{1}+s_{2}+s_{3}+s_{4}-1}{2}-it-\tau \right)\\ & \qquad \times \Gamma(1+\tau-s_{1}-s_{3}) \Gamma(1+\tau-s_{1}-s_{4}) \mathring{g}(\tau, s_{1}) \frac{d\tau}{\pi i}. \end{align*} $$

In a similar fashion, the holomorphic contribution turns into

$$ \begin{align*} \mathscr{L}^{\;\text{hol}} \Psi_{\texttt{s}}^{+}(k) &= i^{k-1} \cos \left(\frac{\pi(s_{3}-s_{4})}{2} \right) \int_{(B)} \frac{\Gamma \left(\frac{k+s_{1}+s_{2}+s_{3}+s_{4}-2}{2}-\tau \right)} {\Gamma \left(\frac{k+2-s_{1}-s_{2}-s_{3}-s_{4}}{2}+\tau \right)}\\ &\quad \times \Gamma(1+\tau-s_{1}-s_{3}) \Gamma(1+\tau-s_{1}-s_{4}) \mathring{g}(\tau, s_{1}) d\tau. \end{align*} $$

To proceed, we focus on the context of the minus sign. In this case, the bound shown in Lemma 3.2 plays a crucial rôle. By definition, the integral transform in question is written as

$$ \begin{align*} \mathscr{L}^{-} \Psi_{\texttt{s}}^{-}(t) = 8i^{-\kappa} \cosh(\pi t) \int_{0}^{\infty} \Psi_{\texttt{s}}^{-}(\eta) K_{2it}(\eta) \frac{d\eta}{\eta}. \end{align*} $$

Inserting equation (3.15), we obtain an absolutely convergent integral, for we have the growth condition

$$ \begin{align*} K_{2it}(\eta) \ll \begin{cases} |\log \eta| & \text{as } \eta \to 0,\\ \exp(-\eta) & \text{as } \eta \to \infty. \end{cases} \end{align*} $$

Substituting Heaviside’s integral formula

(3.25) $$ \begin{align} \int_{0}^{\infty} K_{2v}(\eta) \left(\frac{\eta}{2} \right)^{2s-1} d\eta = \frac{\Gamma(s+v) \Gamma(s-v)}{2} \quad \text{for} \quad \Re(s)> |\Re(v)|, \end{align} $$

we are left with the expression

$$ \begin{align*} \mathscr{L}^{-} \Psi_{\texttt{s}}^{-}(t) &= -i^{-\kappa} \cosh(\pi t) \int_{(B)} \cos \left(\pi \left(s_{1}+\frac{s_{3}+s_{4}}{2}-\tau \right) \right) \Gamma \left(\frac{s_{1}+s_{2}+s_{3}+s_{4}-1}{2}+it-\tau \right)\\ &\quad \times \Gamma \left(\frac{s_{1}+s_{2}+s_{3}+s_{4}-1}{2}-it-\tau \right) \Gamma(1+\tau-s_{1}-s_{3}) \Gamma(1+\tau-s_{1}-s_{4}) \mathring{g}(\tau, s_{1}) \frac{d\tau}{\pi i}. \end{align*} $$

In anticipation of future simplifications of $\mathscr {L}^{\pm } \Psi _{\texttt {s}}^{\pm }(t)$ , we define the the integral transforms

(3.26)

and

(3.27)

along with the original integral transform

(3.28)

Here the contour in the formula for $\Phi _{\texttt {s}}^{+}(\xi )$ is curved to ensure that the poles of the first two gamma factors in the integrand lie to the right of the contour and those of the other factors are on the left of the contour. In addition, the variables $\xi , s_{1}, s_{2}, s_{3}, s_{4}$ are assumed to be such that the path can be drawn. The contour in the definition of $\Phi _{\texttt {s}}^{-}(\xi )$ is chosen in just the same way. On the other hand, the contour in $\Xi $ separates the poles of $\Gamma (\xi +(s_{1}+s_{2}+s_{3}+s_{4}-1)/2-\tau )$ and those of $\Gamma (1+\tau -s_{1}-s_{3}) \Gamma (1+\tau -s_{1}-s_{4}) \mathring {g}(\tau , s_{1})$ to the left and the right of the contour. An explicit formulation of the integral transforms can be executed:

Proposition 3.5. With the notation above, we have that

$$ \begin{align*} \Phi_{\texttt{s}}^{+}(\xi) &= -(-i\xi)^{\kappa} \frac{(2\pi)^{s_{1}-s_{2}}}{2\sin(\pi \xi)} \cos \left(\frac{\pi(s_{3}-s_{4})}{2} \right) \Big\{\Xi_{\texttt{s}}(\xi)-(-1)^{\kappa} \Xi_{\texttt{s}}(-\xi) \Big\},\\ \Phi_{\texttt{s}}^{-}(\xi) &= i^{-\kappa} \frac{(2\pi)^{s_{1}-s_{2}}}{2\sin(\pi \xi)} \left\{\sin \left(\pi \left(\frac{s_{2}-s_{1}}{2}+\xi \right) \right) \Xi_{\texttt{s}}(\xi) - \sin \left(\pi \left(\frac{s_{2}-s_{1}}{2}-\xi \right) \right) \Xi_{\texttt{s}}(-\xi) \right\}, \end{align*} $$

provided the left-hand sides are well-defined. We also have for real t and $\texttt {s} = (s_{1}, s_{2}, s_{3}, s_{4}) \in \mathcal {E}_{4}$ that

$$ \begin{align*} \mathscr{L}^{+} \Psi_{\texttt{s}}^{+}(t) = (2\pi)^{1-s_{1}+s_{2}} \Phi_{\texttt{s}}^{+}(it),\\ \mathscr{L}^{-} \Psi_{\texttt{s}}^{-}(t) = (2\pi)^{1-s_{1}+s_{2}} \Phi_{\texttt{s}}^{-}(it). \end{align*} $$

For integral $k \equiv \kappa \pmod {2}$ and $(s_{1}, s_{2}, s_{3}, s_{4}) \in \mathcal {E}_{4}$ , we have that

$$ \begin{align*} \mathscr{L}^{\;\mathrm{hol}} \Psi_{\texttt{s}}^{+}(k) = 2\pi i^{k} \cos \left(\frac{\pi(s_{3}-s_{4})}{2} \right) \Xi_{\texttt{s}} \left(\frac{k-1}{2} \right). \end{align*} $$

3.5 Deduction of the cubic moment side

We devote this subsection to the derivation of the cubic moment side. The absolute convergence that we would like to check is apparent as far as the double summation over the variables $m, n$ is concerned, since we have the bound in equation (2.22) on the Hecke eigenvalues and $\texttt {s} = (s_{1}, s_{2}, s_{3}, s_{4}) \in \mathcal {E}_{4}$ . Hence, the chief issue is reduced to bounding $\mathscr {L}^{\pm } \Psi _{\texttt {s}}^{\pm }$ , and then Proposition 3.5 reduces our task to the analysis of the function $\Xi $ . If real t and positive integral k tend to infinity, we have uniformly for any compact subset of $\mathcal {E}_{4}$ that

$$ \begin{align*} \Xi_{\texttt{s}}(it) \ll |t|^{-A}, \qquad \Xi_{\texttt{s}} \left(\frac{k-1}{2} \right) \ll k^{-A}. \end{align*} $$

3.5.1 Computation of $\mathcal {J}_{\pm }$

We are able to insert equation (3.21) into equation (3.19) and change the order of summations and integrals as long as we work inside $\mathcal {E}_{4}$ . The replacement of $\psi $ (in the discussion of Section 3.4) with $\psi ^{2}$ is necessary. We now evaluate Rankin–Selberg L-functions involving the divisor function defined as

One then establishes the following lemma:

Lemma 3.7. For $f \in \mathcal {B}_{\kappa }(dq, \psi ^{2})$ and an Eisenstein series $E(z, 1/2+it, f)$ with $f \in \mathcal {B}(\psi _{1}, \psi _{2})$ , we have

$$ \begin{align*} \mathcal{T}(s, u; f; \psi) &= \frac{L(s, f \otimes \overline{\psi}) L(s-u, f \otimes \overline{\psi})}{\zeta^{q}(2s-u)},\\ \mathcal{T}(s, u; E; \psi) &= \frac{L(s+it, \overline{\psi} \psi_{2}) L(s-it, \overline{\psi} \psi_{1}) L(s-u+it, \overline{\psi} \psi_{2}) L(s-u-it, \overline{\psi} \psi_{1})}{\zeta^{q}(2s-u)}. \end{align*} $$

Remark 3.8. The level $dq$ in Lemma 3.7 can be replaced with any positive integer that is divided by the modulus q of the central character.

Proof. For the first assertion, we use the multiplicativity relation in equation (2.21) for the Hecke eigenvalues, getting

$$ \begin{align*} \mathcal{T}(s, u; f; \psi) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{\overline{\psi}(mn) \lambda_{f}(mn)}{m^{s} n^{s-u}} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{\overline{\psi}(mn)}{m^{s} n^{s-u}} \sum_{d \mid (m, n)} \mu(d) \psi(d)^{2} \lambda_{f} \left(\frac{m}{d} \right) \lambda_{f} \left(\frac{n}{d} \right). \end{align*} $$

Upon exchanging the order of summations, the right-hand side equals

$$ \begin{align*} \sum_{(d, q) = 1} \frac{\mu(d)}{d^{2s-u}} \sum_{m = 1}^{\infty} \frac{\overline{\psi}(m) \lambda_{f}(m)}{m^{s}} \sum_{n = 1}^{\infty} \frac{\overline{\psi}(n) \lambda_{f}(n)}{n^{s-u}} = \frac{L(s, f \otimes \overline{\psi}) L(s-w, f \otimes \overline{\psi})}{\zeta^{q}(2s-u)}. \end{align*} $$

In order to establish the second assertion, we make use of Lemma 2.5 to derive

$$ \begin{align*} \mathcal{T}(s, u; E; \psi) & = \sum_{m = 1}^{\infty} \frac{\overline{\psi}(m) \sigma_{u}(m)}{m^{s}} \sum_{a \mid m} \psi_{1}(a) \psi_{2} \left(\frac{m}{a} \right) \left(\frac{a^{2}}{m} \right)^{it}\\ & = \sum_{(c, q) = 1} \frac{\mu(c)}{c^{2s-u}} \sum_{a = 1}^{\infty} \frac{\overline{\psi}(a) \psi_{1}(a) \sigma_{w}(a)}{a^{s-it}} \sum_{m = 1}^{\infty} \frac{\overline{\psi}(m) \psi_{2}(m) \sigma_{u}(m)}{m^{s+it}}\\ & = \frac{L(s+it, \overline{\psi} \psi_{2}) L(s-it, \overline{\psi} \psi_{1}) L(s-u+it, \overline{\psi} \psi_{2}) L(s-u-it, \overline{\psi} \psi_{1})}{\zeta^{q}(2s-u)}. \end{align*} $$

This concludes the proof of Lemma 3.7.

We proceed to compute

by virtue of Lemma 3.7, obtaining

$$ \begin{align*} \sum_{n = 1}^{\infty} \frac{\overline{\psi}(n) \sigma_{s_{4}-s_{3}}(n) \rho_{f}(\pm n)}{n^{(1-s_{1}+s_{2}-s_{3}+s_{4})/2}} = \epsilon_{f}^{(1 \mp 1)/2} \rho_{f}(1) \frac{L \left(\frac{1-s_{1}+s_{2}+s_{3}-s_{4}}{2}, f \otimes \overline{\psi} \right) L \left(\frac{1-s_{1}+s_{2}-s_{3}+s_{4}}{2}, f \otimes \overline{\psi} \right)}{\zeta^{q}(1-s_{1}+s_{2})}, \end{align*} $$

where $f \in \mathcal {B}_{\kappa }(dq, \psi ^{2})$ . There is beauty in the above quotient since we have $\zeta ^{q}(1-s_{1}+s_{2})^{-1}$ , which completely cancels out with the term $\zeta ^{q}(1-s_{1}+s_{2})$ appearing in the expression for $\mathcal {J}_{\pm }$ in Proposition 3.3. One can calculate the Dirichlet series for the sum over m in a similar manner, obtaining

$$ \begin{align*} \sum_{m = 1}^{\infty} \frac{\psi(m) \overline{\rho_{f}(m)}}{m^{(s_{1}+s_{2}+s_{3}+s_{4}-1)/2}} = \overline{\rho_{f}(1)} L \left(\frac{s_{1}+s_{2}+s_{3}+s_{4}-1}{2}, f \otimes \overline{\psi} \right), \end{align*} $$

where $f \in \mathcal {B}_{\kappa }(dq, \psi ^{2})$ . We obtain the central L-value $L(1/2, f \otimes \overline {\psi })^{3}$ if we take the limit $(s_{1}, s_{2}, s_{3}, s_{4}) \mapsto (1/2, 1/2, 1/2, 1/2)$ . A similar procedure can be applied to the Eisenstein contribution to deduce the six Dirichlet L-functions in equation (1.4). The eventual form of the cubic moment side looks like

with

Here we set $\mathcal {H}_{\pm }(\chi , \psi ) = \psi (\mp 1) \tau (\psi ) \tau (\overline {\chi \psi }) J(\psi , \psi )$ and used the identity $\varphi (dq) = d \varphi (q)$ for q prime.

3.5.2 Computation of $\mathcal {E}_{\pm }$

We initially handle

. One can rewrite the direct sum decomposition in equation (2.20) in order to imitate the formulation of Blomer–Milićević [Reference Blomer and Milićeić14], namely

$$ \begin{align*} \mathcal{B}_{0}(\Gamma_{0}(r^{2})) = \bigsqcup_{r_{1} r_{2} = r^{2}} \bigsqcup_{f \in \mathcal{B}_{0}^{\ast}(\Gamma_{0}(r_{1}))} \bigsqcup_{d \mid r_{2}} \iota_{d} f \cdot \mathbb{C}, \end{align*} $$

where the first two sums are orthogonal, but the last one is not orthogonal and needs to be orthogonalised by Gram–Schmidt. Here we also define

. By [Reference Blomer and Milićeić14, Lemma 9], the set of functions

is an orthonormal basis of the space $\bigsqcup _{d \mid r_{2}} \iota _{d} f \cdot \mathbb {C}$ with $f \in \mathcal {B}_{0}^{\ast }(\Gamma _{0}(r_{1}))$ . Then the Fourier coefficients are

$$ \begin{align*} \rho_{f^{(g)} \mathfrak{a}}(n) = \sum_{d \mid g} \xi_{g}(d) d^{1/2} \rho_{f \mathfrak{a}} \left(\frac{n}{d} \right) \end{align*} $$

with the convention that $\rho _{f \mathfrak {a}}(x) = 0$ for $x \not \in \mathbb {Z}$ . We calculate the sum over n in equation (3.20) as

$$ \begin{align*} \sum_{n = 1}^{\infty} \frac{\psi(n) \sigma_{s_{4}-s_{3}}(n) \rho_{f 0}(\pm n/d)}{n^{(1-s_{1}+s_{2}-s_{3}+s_{4})/2}} = \delta_{d = 1} \epsilon_{f}^{(1 \mp 1)/2} \omega_{f} \rho_{f}(1) \frac{L \left(\!\frac{1-s_{1}+s_{2}+s_{3}-s_{4}}{2}, f \otimes \psi \!\right) L \left(\!\frac{1-s_{1}+s_{2}-s_{3}+s_{4}}{2}, f \otimes \psi \!\right)}{\zeta^{r}(1-s_{1}+s_{2})}, \end{align*} $$

where $\omega _{f}$ stands for the root number for a newform $f \in \mathcal {B}_{0}^{\ast }(\Gamma _{0}(r_{1}))$ . We denote by $\psi $ a Dirichlet character modulo r induced by the primitive character $\psi ^{\ast }$ modulo $r^{\ast }$ . We regard the character modulo 1 as primitive. Since the central character is trivial and the Hecke eigenvalues are real (Remark 2.10), we use Lemma 2.1 to infer that

$$ \begin{align*} \sum_{m = 1}^{\infty} \frac{\tau(\psi, m) \overline{\rho_{f}(m/d)}}{m^{(s_{1}+s_{2}+s_{3}+s_{4}-1)/2}} = \frac{\overline{\rho_{f}(1)} \tau(\psi^{\ast}) \psi^{\ast}(d)}{d^{(s_{1}+s_{2}+s_{3}+s_{4}-1)/2}} \Sigma_{r/r^{\ast}}(s_{1}, s_{2}, s_{3}, s_{4}; \psi) L \left(\frac{s_{1}+s_{2}+s_{3}+s_{4}-1}{2}, f \otimes \psi^{\ast} \right), \end{align*} $$

where $\Sigma _{q}(s_{1}, s_{2}, s_{3}, s_{4}; \psi )$ is the multiplicative function

$$ \begin{align*} \sum_{c \mid q} \mu \left(\frac{q}{c} \right) \psi^{\ast} \left(\frac{q}{c} \right) c^{(3-s_{1}-s_{2}-s_{3}-s_{4})/2} \sum_{e \mid c} \mu(e) \psi_{0}(e) \psi^{\ast}(e) e^{(1-s_{1}-s_{2}-s_{3}-s_{4})/2} \lambda_{f} \left(\frac{c}{e} \right) \end{align*} $$

and $\psi _{0}$ is the principal character modulo $r_{1}$ . On the other hand, we obtain via elementary calculations that

(3.29) $$ \begin{align} \sum_{n \mid r^{\infty}} \frac{\tau(\psi, n) \sigma_{w}(n) \lambda_{f}(n)}{n^{s}} = \tau(\psi^{\ast}) \sum_{c \mid r/r^{\ast}} c \psi^{\ast} \left(\frac{r}{cr^{\ast}} \right) \mu \left(\frac{r}{cr^{\ast}} \right) \sum_{n \mid r^{\infty}} \frac{\psi^{\ast}(n) \sigma_{w}(nc) \lambda_{f}(nc)}{(nc)^{s}}, \end{align} $$

where $s = (1-s_{1}+s_{2}-s_{3}+s_{4})/2$ and $w = s_{4}-s_{3}$ . We hence observe that the left-hand side of equation (3.29) equals $\tau (\psi ^{\ast })$ when $\psi $ is the quadratic character modulo q or the character modulo $1$ . The problem occurs if $\psi $ is the principal character modulo a prime q. In this case, a brute force computation gives

$$ \begin{align*} \sum_{c \mid q} c^{1-s} \mu \left(\frac{q}{c} \right) \sum_{n \mid q^{\infty}} \frac{\sigma_{w}(nc) \lambda_{f}(nc)}{n^{s}} = \varphi(q) \left(1-\frac{\lambda_{f}(q)}{q^{s}} \right)^{-1} \left(1-\frac{\lambda_{f}(q)}{q^{s-w}} \right)^{-1}-q. \end{align*} $$

We then conclude that

$$ \begin{align*} \sum_{n \mid r^{\infty}} \frac{\tau(\psi, n) \sigma_{w}(n) \lambda_{f}(n)}{n^{(1-s_{1}+s_{2}-s_{3}+s_{4})/2}} = \tau(\psi^{\ast}) \Pi_{r/r^{\ast}}(s_{1}, s_{2}, s_{3}, s_{4}; \psi), \end{align*} $$

where $\Pi _{q}(s_{1}, s_{2}, s_{3}, s_{4}; \psi )$ is the multiplicative function

$$ \begin{align*} q \sum_{c \mid q} \mu \left(\frac{q}{c} \right) \prod_{p \mid c} \left(1-\frac{\psi^{\ast}(p)}{p} \right) \left(1-\frac{\psi^{\ast}(p) \lambda_{f}(p)}{p^{(1-s_{1}+s_{2}+s_{3}-s_{4})/2}} \right)^{-1} \left(1-\frac{\psi^{\ast}(p) \lambda_{f}(p)}{p^{(1-s_{1}+s_{2}-s_{3}+s_{4})/2}} \right)^{-1}. \end{align*} $$

This reasoning proceeds verbatim for the Eisenstein and holomorphic spectra, giving similar expressions (see [Reference Young70] for an extension of Weisinger’s theory and newform Eisenstein series). Thus we are left with

where

where we define $\mathcal {G}_{\pm }(\chi , \psi ) = \psi (\mp 1) \tau (\psi ) \tau (\psi ^{\ast })^{2} \tau (\overline {\chi } \psi )$ and

$$ \begin{align*} \Omega_{r_{1}, r_{2}}(s_{1}, s_{2}, s_{3}, s_{4}; f; \psi) = \frac{\varphi(r_{1})}{r_{1}^{2}} \sum_{g \mid r_{2}} \xi_{g}(1) &\sum_{d \mid g} \xi_{g}(d) \psi^{\ast}(d) d^{(2-s_{1}-s_{2}-s_{3}-s_{4})/2}\\ &\qquad \times \Sigma_{r/r^{\ast}}(s_{1}, s_{2}, s_{3}, s_{4}; \psi) \Pi_{r/r^{\ast}}(s_{1}, s_{2}, s_{3}, s_{4}; \psi). \end{align*} $$

These terms are complicated, but they are the same size as the main contributions $\mathcal {J}_{\pm }$ in estimations. Thus one can ignore them in the process of using Motohashi’s formula to deduce some moment bounds.

3.6 Endgame: analytic continuation

The aim of this subsection is to show that the spectral decomposition obtained in the preceding subsection can be continued to the whole complex plane $\mathbb {C}^{4}$ and hence to conclude the proof of Theorem 1.1. The main step is identical to that of Motohashi’s monograph [Reference Motohashi55], and we stress the following lemmata:

Lemma 3.9 (Motohashi [Reference Motohashi55, Lemma 4.7])

The function $\Xi _{\texttt {s}}(\xi )$ is meromorphic in the domain

$$ \begin{align*} \mathcal{B}_{4}^{\ast} = \{\xi: \Re(\xi)> -cA \} \times \mathcal{B}_{4} \end{align*} $$

for a fixed small constant $c> 0$ and holomorphic in $\mathcal {B}_{4}^{\ast } \setminus \mathcal {N}$ , where

$$ \begin{align*} \mathcal{N} = \left\{ (\xi, s_{1}, s_{2}, s_{3}, s_{4})\ : \begin{array}{l} \text{at least one of } \xi+\dfrac{s_{1}+s_{2}+s_{3}+s_{4}-1}{2}, \xi+\dfrac{1-s_{1}+s_{2}+s_{3}-s_{4}}{2}, \\ \ \xi+\dfrac{1-s_{1}+s_{2}-s_{3}+s_{4}}{2} \text{ equals a non-positive integer} \end{array} \right\}. \end{align*} $$

Moreover, if $|\xi |$ tends to infinity in any fixed vertical or horizontal strips while satisfying $\Re (\xi )> -cA$ , then we have uniformly in $\mathcal {B}_{4}$ that

(3.30) $$ \begin{align} \Xi_{\texttt{s}}(\xi) \ll |\xi|^{-cA}. \end{align} $$

Lemma 3.10 (Motohashi [Reference Motohashi55, Lemma 4.8])

If $(\xi , s_{1}, s_{2}, s_{3}, s_{4})$ is such that the path in equation (3.28) can be drawn in a vertical strip contained in the half plane $\Re (\tau )> 0$ , then we have that

$$ \begin{align*} \Xi_{\texttt{s}}(\xi) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\gamma)} \int_{0}^{\infty} y^{\xi+(s_{1}+s_{2}+s_{3}+s_{4}-3)/2} G(y, s_{1}) \ {_{2}}F_{1}(\alpha, \beta; \gamma; -y) dy, \end{align*} $$

where $G(y, s)$ is defined in equation (3.4) and ${_{2}}F_{1}(\alpha , \beta; \gamma; y)$ is the hypergeometric function with

$$ \begin{align*} \alpha = \xi+\frac{1-s_{1}+s_{2}+s_{3}-s_{4}}{2}, \qquad \beta = \xi+\frac{1-s_{1}+s_{2}-s_{3}+s_{4}}{2}, \qquad \gamma = 1+2\xi. \end{align*} $$

We assume that $(s_{1}, s_{2}, s_{3}, s_{4}) \in \mathcal {B}_{4}$ and examine the consequence of Lemma 3.9 for the contribution from Maaß forms. If $t_{f} \geqslant 3B$ , then $\Xi _{\texttt {s}}(\pm it_{f})$ is holomorphic and $\mathrm {O}(t_{f}^{-cA})$ uniformly in $\mathcal {B}_{4}$ with an absolute constant c. Hence via Proposition 3.5, the function $\mathscr {L}^{+} \Psi _{\texttt {s}}^{+}(t_{f})$ is holomorphic and of exponential decay with respect to $t_{f}$ uniformly in $\mathcal {B}_{4}$ . This indicates that exists as a meromorphic function inside $\mathcal {B}_{4}$ . The same observation holds for $\mathcal {J}_{+}^{\;\text {hol}}$ . As for the function , we essentially need equation (3.30). From Proposition 3.5, we have $\mathscr {L}^{-} \Psi _{\texttt {s}}^{-}(t_{f}) = \mathrm {O}(t_{f}^{-cA})$ uniformly in $\mathcal {B}_{4}$ provided $t_{f} \geqslant 3B$ . Hence is meromorphic inside $\mathcal {B}_{4}$ . This discussion works for and $\mathcal {E}_{+}^{\;\text {hol}}$ as well.

Thus it remains to contemplate $\mathcal {J}_{\pm }^{\;\text {Eis}}$ and $\mathcal {E}_{\pm }^{\;\text {Eis}}$ . To this end, we assume first that $\texttt {s} = (s_{1}, s_{2}, s_{3}, s_{4}) \in \mathcal {E}_{4}$ and set

$$ \begin{align*} \mathcal{J}^{\;\text{Eis}}(\texttt{s}; g; \chi) = \mathcal{J}_{+}^{\;\text{Eis}}(\texttt{s}; g; \chi) + \mathcal{J}_{-}^{\;\text{Eis}}(\texttt{s}; g; \chi). \end{align*} $$

Using Proposition 3.5, one derives

$$ \begin{align*} \Phi_{\texttt{s}}^{+}(it) &= -\frac{(2\pi)^{s_{1}-s_{2}}}{2i \sinh(\pi t)} \cos \left(\frac{\pi(s_{3}-s_{4})}{2} \right) \Big\{\Xi_{\texttt{s}}(it)-\Xi_{\texttt{s}}(-it) \Big\},\\ \Phi_{\texttt{s}}^{-}(it) &= \frac{(2\pi)^{s_{1}-s_{2}}}{2i \sinh(\pi t)} \left\{\sin \left(\pi \left(\frac{s_{2}-s_{1}}{2}+it \right) \right) \Xi_{\texttt{s}}(it)-\sin \left(\pi \left(\frac{s_{2}-s_{1}}{2}-it \right) \right) \Xi_{\texttt{s}}(-it) \right\}. \end{align*} $$

This yields an expression of $\mathcal {J}^{\;\text {Eis}}(\texttt {s}; g; \chi )$ in terms of $\Xi _{\texttt {s}}(it)$ . We need to shift the contour in the same way as on pages 170–171 of Motohashi’s monograph [Reference Motohashi55]. This is because $\mathcal {S}_{f}(t; s_{1}, s_{2}, s_{3}, s_{4})$ has poles when $\psi _{1} = \psi _{2} = \psi $ . The same procedure applies to the term $\mathcal {E}_{\pm }^{\;\text {Eis}}$ coming from the contribution of the principal character and quadratic character. Gathering these observations, we have proven the following lemma:

This concludes the proof of Theorem 1.1.

4.1 Proofs of Corollaries 1.3 and 1.4

It suffices to specialise the test function in Theorem 1.1 as

(4.1) $$ \begin{align} g(t) = \frac{1}{\sqrt{\pi} H} \exp \left(-\left(\frac{t - T}{H} \right)^{2} \right) \end{align} $$

with the parameter H at one’s disposal. We assume that

$$ \begin{align*} T^{1/2} \leqslant H \leqslant T(\log T)^{-1}. \end{align*} $$

Motohashi [Reference Motohashi55, (5.1.40)–(5.1.42)] evaluated the corresponding integral transform $\Xi $ , and his expression can be used in our context. We then derive an asymptotic formula almost identical to [Reference Motohashi55, Theorem 5.1]. Upon estimating the spectral sum by absolute values, an oscillatory component in the summand evanishes, and we obtain the bound

(4.2) $$ \begin{align} \int_{T}^{T+H} \left|L \left(\frac{1}{2}+it, \chi \right) \right|^{4} dt \ll_{\epsilon} H^{1+\epsilon} q^{\epsilon} + \frac{H^{3/2}}{T} q^{-2+\epsilon} \sum_{\psi\! \pmod q} \sum_{\substack{f \in \mathcal{B}_{\kappa}^{\ast}(q^{2}, \psi^{2}) \\ t_{f} \ll T/H}} L \left(\frac{1}{2}, f \otimes \overline{\psi} \right)^{3}. \end{align} $$

The truncation of the sum is justified since the Taylor expansion of $\exp (-(Ht_{f}/T)^{2}/4)$ in [Reference Motohashi55, (5.1.44)] implies that the rest of the sum is smaller than the main term in equation (4.2). Notice that the central L-values in equation (4.2) are nonnegative, and it thus follows from the work of Petrow–Young [Reference Petrow and Young63, Theorems 1.2 & 1.3] that

$$ \begin{align*} H^{1+\epsilon} q^{\epsilon}+\frac{H^{3/2}}{T} q^{-1+\epsilon} \left(\frac{qT}{H} \right)^{2+\epsilon} \ll H^{1+\epsilon} q^{\epsilon}+\left(\frac{qT}{\sqrt{H}} \right)^{1+\epsilon}. \end{align*} $$

Optimising $H = (qT)^{2/3}$ concludes the proof of Corollaries 1.3 and 1.4.

4.2 Proofs of Corollaries 1.5 and 1.6

This section is devoted to proving the twelfth moment bound. We follow the argument of [Reference Ivić and Motohashi34] verbatim, obtaining the expressionFootnote ⁶

$$ \begin{align*} \sum_{r = 1}^{R} \int_{t_{r}}^{t_{r}+T_{0}} \left|L \left(\frac{1}{2}+it, \chi \right) \right|^{4} dt &\ll_{\epsilon} T_{0} q^{-2+\epsilon} \sum_{\psi \pmod{q} } \sum_{\substack{f \in \mathcal{B}_{\kappa}^{\ast}(q^{2}, \psi^{2}) \\ t_{f} \leqslant TT_{0}^{-1} \sqrt{\log T}}} t_{f}^{-1/2} \frac{L(1/2, f \otimes \overline{\psi})^{3}}{L(1, \mathrm{Ad}^{2} f)}\\ & \qquad \times \sum_{r = 1}^{R} t_{r}^{-1/2} \exp \left(-\left(\frac{T_{0} t_{f}}{2t_{r}} \right)^{2} \right) \sin \left(t_{f} \log \frac{t_{f}}{4et_{r}} \right)+RT_{0}(qT)^{\epsilon}. \end{align*} $$

For technical reasons, it is convenient to remove the exponential factor in the last sum over r by partial summation. In doing this, we obtain an error term of $\mathrm {O}_{\epsilon }(RT^{\epsilon } T_{0}^{-1})$ . Then we must majorise

where

The Cauchy–Schwarz inequality leads one to

$$ \begin{align*} S_{K} &\leqslant \left(\sum_{\psi \pmod{q} } \sum_{\substack{f \in \mathcal{B}_{\kappa}^{\ast}(q^{2}, \psi^{2}) \\ K \leqslant t_{f} \leqslant 2K}} t_{f}^{-1} \frac{L(1/2, f \otimes \overline{\psi})^{4}}{L(1, \mathrm{Ad}^{2} f)} \right)^{1/2}\\ & \times \left(\sum_{\psi \pmod{q} } \sum_{\substack{f \in \mathcal{B}_{\kappa}^{\ast}(q^{2}, \psi^{2}) \\ K \leqslant t_{f} \leqslant 2K}} \frac{L(1/2, f \otimes \overline{\psi})^{2}}{L(1, \mathrm{Ad}^{2} f)} \left|\sum_{r = 1}^{R} t_{r}^{-1/2-it_{f}} \right|^{2} \right)^{1/2}. \end{align*} $$

Applying the bound of Petrow–Young [Reference Petrow and Young63, Theorem 7.6], the fourth moment is bounded as

$$ \begin{align*} \sum_{\psi \pmod{q} } \sum_{\substack{f \in \mathcal{B}_{\kappa}^{\ast}(q^{2}, \psi^{2}) \\ K \leqslant t_{f} \leqslant 2K}} t_{f}^{-1} \frac{L(1/2, f \otimes \overline{\psi})^{4}}{L(1, \mathrm{Ad}^{2} f)} \ll_{\epsilon} q^{3+\epsilon} K^{1+\epsilon}. \end{align*} $$

The spectral large sieve in the form involving the square of twisted automorphic L-functions yields

$$ \begin{align*} \sum_{\psi \pmod{q} } \sum_{\substack{f \in \mathcal{B}_{\kappa}^{\ast}(q^{2}, \psi^{2}) \\ K \leqslant t_{f} \leqslant 2K}} \frac{L(1/2, f \otimes \overline{\psi})^{2}}{L(1, \mathrm{Ad}^{2} f)} \left|\sum_{r = 1}^{R} t_{r}^{-1/2-it_{f}} \right|^{2} \ll_{\epsilon} q^{3+\epsilon} RKT^{\epsilon} T_{0}^{-1}, \end{align*} $$

since $K \ll TT_{0}^{-1} \sqrt {\log T}$ . It follows that

$$ \begin{align*} S_{K} \ll_{\epsilon} q^{3+\epsilon} R^{1/2} K^{1+\epsilon} T^{\epsilon} T_{0}^{-1/2} \end{align*} $$

and the summation over K eventually gives

$$ \begin{align*} \sum_{r = 1}^{R} \int_{t_{r}}^{t_{r}+T_{0}} \left|L \left(\frac{1}{2}+it, \chi \right) \right|^{4} dt \ll_{\epsilon} \left(RT_{0}+qT \sqrt{\frac{R}{T_{0}}} \right)(qT)^{\epsilon}. \end{align*} $$

This establishes Corollary 1.5. Corollary 1.6 follows from Jutila’s trick described in [Reference Jutila42, §6].