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1 Introduction↩︎

The study of the value distribution of automorphic forms is a central problem in analytic number theory and arithmetic quantum chaos. Let \(\mathbb{H}=\{z=x+iy:x\in\mathbb{R},\;y>0\}\) denote the upper half plane, and let \(\mathrm{d}\mu z= \mathrm{d}x\mathrm{d}y/y^2\) be the hyperbolic measure. Let \(\Gamma=\operatorname{SL}_2(\mathbb{Z})\) be the modular group. Let \(k\geq12\) be an even integer, and let \(H_k\) be a Hecke basis for the space \(S_k\) of all holomorphic cusp forms of weight \(k\) for \(\Gamma\). For \(f\in H_k\), we normalize it so that \(\langle f,f\rangle_k := \int_{\Gamma\backslash \mathbb{H}} y^k |f(z)|^2 \mathrm{d}\mu z =\operatorname{vol}(\Gamma\backslash \mathbb{H})=\pi/3\).

In the \(L^2\)-setting, we have the holomorphic analog of the quantum unique ergodicity (hQUE) conjecture of Rudnick and Sarnak [1]. In 2010, Holowinsky and Soundararajan [2] proved hQUE, confirming the equidistribution of the mass of \(f\). Specifically, they proved that \[\frac{1}{\operatorname{vol}(\Omega)} \int_{\Omega} y^{k} |f(z)|^{2} \mathrm{d}\mu z = 1 + o(1), \quad \textrm{as k\rightarrow \infty,}\] for any fixed compact domain \(\Omega\) of \(\Gamma\backslash\mathbb{H}\) with hyperbolic measure zero boundary \(\partial\Omega\).

In 2013, Blomer, Khan, and Young [3] studied the \(L^4\)-norm of \(f\), proving that \[\label{eqn:L944} \int_{\Gamma\backslash \mathbb{H}} y^{2k}|f(z)|^{4} \mathrm{d}\mu z = O( k^{1/3+\varepsilon}),\tag{1}\] for any \(\varepsilon>0\). They conjectured that \[\frac{1}{\operatorname{vol}(\Gamma\backslash \mathbb{H})} \int_{\Gamma\backslash \mathbb{H}} y^{2k}|f(z)|^{4} \mathrm{d}\mu z=2+o(1), \quad \text{as k\rightarrow\infty.}\] Assuming the generalized Riemann Hypothesis (GRH), Zenz [4] recently proved \[\int_{\Gamma\backslash \mathbb{H}} y^{2k}|f(z)|^{4} \mathrm{d}\mu z = O(1).\] Blomer, Khan, and Young [3] also showed that for \(p>6\), we have \[\int_{\Gamma\backslash \mathbb{H}} |y^{k/2}f(z)|^p \mathrm{d}\mu z \gg k^{p/4-3/2-\varepsilon},\] which extended the sup-norm result of Xia [5], namely \(\max_{z\in\mathbb{H}} |y^{k/2}f(z)| = k^{1/4+o(1)}\).

In general, it is natural to conjecture [6] that for any \(a\in \mathbb{N}\), we have \[\frac{1}{\operatorname{vol}(\Omega)} \int_{\Omega} y^{ak} |f(z)|^{2a} \mathrm{d}\mu z = a! + o(1), \quad \textrm{ as k\rightarrow \infty,}\] for any fixed compact domain \(\Omega \subset \Gamma\backslash\mathbb{H}\) as above. See, for instance, [7] for the expected value of the \(L^p\)-norms of random cusp forms.

Recently, we studied the joint distribution of Hecke eigenforms in the large-weight limit. Let \(f\in H_k\) and \(g\in H_\ell\) and assume \(\langle f,g\rangle=0\) if \(k=\ell\). Denote \(F_k(z):=y^{k/2}f(z)\) and \(G_\ell(z):=y^{\ell/2}g(z)\). We know that \(|F_k(z)|\) and \(|G_\ell(z)|\) are \(\Gamma\) invariant. We will focus on the joint mass distribution of \(|F_k(z)|\) and \(|G_\ell(z)|\), especially the mixed \((2,2)\)-moment \[\langle |F_k|^2, |G_\ell|^2 \rangle := \int_{\Gamma\backslash\mathbb{H}} |F_k(z)|^2 |G_\ell(z)|^2 \mathrm{d}\mu z .\] In [6], we conjectured that orthogonal Hecke eigenforms are statistically independent. In particular, we expected that \[\label{eqn:mixed22} \frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})} \langle |F_k|^2, |G_\ell|^2 \rangle = 1 +o(1),\tag{2}\] as \(\max(k,\ell)\rightarrow\infty\). We established 2 under the assumptions of the generalized Riemann Hypothesis (GRH) and the generalized Ramanujan conjecture (GRC).

In the present paper, we seek unconditional results in this direction. We consider certain expectation and variance of the mixed \((2,2)\)-moment of two Hecke eigenforms of distinct weights. As a consequence, we derive asymptotic formulas for a first moment of triple product \(L\)-functions, leading to new nonvanishing results for these \(L\)-functions.

1.1 The variance↩︎

We prove that the variance of the mixed \((2,2)\)-moment of two Hecke eigenforms vanishes asymptotically. Our main result on the variance is the following theorem.

For even integer \(k\geq12\), we have \(|H_k|=k/12+O(1)\). Theorem 1 implies that 2 holds for all but \(O(K^{2-\varepsilon})\) forms \(f\in \cup_{K<k\leq 2K}H_k\), provided \(g\in H_\ell\) with \(\ell \leq K^{\delta_2-\varepsilon}\). By the Cauchy–Schwarz inequality, we deduce the following asymptotic formula for the expectation.

1.2 An application to the triple product \(L\)-functions↩︎

Let \(f\in H_k\) with the \(n\)-th Hecke eigenvalue \(\lambda_f(n)\). Denote by \(\alpha_{f}(p)\), \(\alpha_{f}(p)^{-1}\) the Satake parameters of \(f\) at a prime \(p\), so that \(\lambda_f(p)=\alpha_f(p)+\alpha_f(p)^{-1}\). The symmetric square \(L\)-function of \(f\) is defined by \[L(s,\operatorname{sym}^2 f) = \zeta(2s) \sum_{n\geq1} \frac{\lambda_f(n^2)}{n^s}, \quad \operatorname{Re}(s)>1,\] which admits analytic continuation to the entire complex plane, and satisfies \(L(1,\operatorname{sym}^2f)\neq0\).

Let \(f\in H_k\), \(g\in H_\ell\) and \(h\in H_{k+\ell}\). The triple product \(L\)-function of \(f,g,h\) is defined by \[L(s,f\times g\times h) = \prod_{p} \prod_{a=\pm1}\prod_{b=\pm1}\prod_{c=\pm1} \left(1-\frac{\alpha_{f}(p)^{a}\alpha_{g}(p)^{b}\alpha_{h}(p)^{c}}{p^s}\right)^{-1}, \quad \operatorname{Re}(s)>1.\] It admits an analytic continuation to the entire complex plane. By Watson’s formula we know \(|\langle fg , h\rangle_{k+\ell}|^2\) is related to the triple product \(L\)-values \(L(1/2,f\times g\times h)\). Note that \(\langle |F_k|^2, |G_\ell|^2 \rangle = \langle fg,fg\rangle_{k+\ell}\). By Parseval’s identity, we have \[\label{eqn:60f2g26261L-values} \frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})} \langle |F_k|^2, |G_\ell|^2 \rangle = \frac{2\pi^2}{k+\ell-1} \sum_{h\in H_{k+\ell}} \frac{ L(1/2,f\times g\times h)}{L(1,\operatorname{sym}^2 h)} \frac{\zeta(2)}{2 L(1,\operatorname{sym}^2 f) L(1,\operatorname{sym}^2 g)}.\tag{3}\] Unconditionally, Blomer, Khan, and Young [3] proved the following nontrivial upper bound \[O((k\ell)^{1/6+\varepsilon}).\] Under GRH, by a similar argument as in Zenz [4], one may prove a sharp upper bound \[O(L(1,\operatorname{sym}^2 f) L(1,\operatorname{sym}^2 g)).\]

As a consequence of Theorem 1, we have the following result for the first moment of \(L(1/2,f\times g\times h)\).

As an immediate corollary, we have the following nonvanishing result for the triple product central \(L\)-values.

1.3 Moments of \(L\)-functions↩︎

To prove Theorem 1, we will use various moments of \(L\)-functions. The symmetric square lift \(\operatorname{sym}^2 f\) of \(f\in H_k\) is a \(\operatorname{GL}_3\) automorphic form. The \((m,n)\)-th Fourier coefficient \(A(m,n)\) of \(\operatorname{sym}^2 f\) is given by \[\label{eqn:A40m44n41} A(m,n) = \sum_{d\mid (m,n)} \mu(d) A(m/d,1)A(1,n/d),\tag{4}\] and \[A(n,1)=A(1,n)= \sum_{a^2b=n}\lambda_f(b^2).\] Here \(\mu\) is the Möbius function.

Let \(\phi\) be a Hecke–Maass cusp form for \(\Gamma\), with the \(n\)-th Hecke eigenvalue \(\lambda_\phi(n)\) and the spectral parameter \(t_\phi\). The Rankin–Selberg \(L\)-function for \(\operatorname{sym}^2 f\) and \(\phi\) is defined by \[\label{eqn:RS3422} L(s,\operatorname{sym}^2 f\times \phi) = \sum_{m,n\geq1} \frac{A(m,n)\lambda_\phi(n)}{(m^2n)^s}, \quad \operatorname{Re}(s)>1.\tag{5}\] It has an analytic continuation to the whole complex plane. We have the following estimate of the first moment of \(L\)-functions.

This extends a result of Luo–Sarnak [9], who treated the case of fixed \(t_\phi\).

Analogously, we will need the following estimate for the second moment of the symmetric square \(L\)-functions.

We also require an upper bound for a mixed moment of \(L\)-functions. Interestingly, this can be deduced from \(L^4\)-norm bounds for Hecke eigenforms.

1.4 Plan for this paper↩︎

The rest of this paper is organized as follows. In §2, we give some lemmas on \(L\)-functions and sums of Fourier coefficients. In §3, we use the spectral method and Watson’s formula to prove Theorem 1. In §4, we prove theorems on moments of \(L\)-functions.

Notation. Throughout the paper, \(\varepsilon\) is an arbitrarily small positive number; all of them may be different at each occurrence. As usual, \(e(x)=e^{2\pi i x}\). We use \(y\asymp Y\) to mean that \(c_1 Y\leq |y|\leq c_2 Y\) for some positive constants \(c_1\) and \(c_2\). The symbol \(\ll_{a,b}\) denotes that the implied constant depends at most on \(a\) and \(b\).

2 Preliminaries↩︎

2.1 \(L\)-functions↩︎

Let \(f\in H_k\) and \(\phi\) an even Hecke–Maass cusp form for \(\Gamma\). The functional equation of the symmetric square \(L\)-function \(L(s,\operatorname{sym}^2 f)\) is \[\Lambda(s,\operatorname{sym}^2 f) = \Lambda(1-s,\operatorname{sym}^2 f),\] where the completed \(L\)-function is defined by \[\Lambda(s,\operatorname{sym}^2 f) := L_\infty(s,\operatorname{sym}^2 f) L(s,\operatorname{sym}^2 f),\] with \[L_\infty(s,\operatorname{sym}^2 f) := \pi^{-3s/3} \Gamma\left(\frac{s+1}{2} \right) \Gamma\left(\frac{s+k-1}{2} \right) \Gamma\left(\frac{s+k}{2} \right) .\] The functional equation of the Rankin–Selberg \(L\)-function \(L(s,\operatorname{sym}^2 f\times \phi)\) is \[\Lambda(s,\operatorname{sym}^2 f\times \phi) = \Lambda(1-s,\operatorname{sym}^2 f\times \phi),\] where the completed \(L\)-function is defined by \[\Lambda(s,\operatorname{sym}^2 f\times \phi) := L_\infty(s,\operatorname{sym}^2 f\times \phi) L(s,\operatorname{sym}^2 f\times \phi),\] with \[L_\infty(s,\operatorname{sym}^2 f\times \phi) := \pi^{-3s}\prod_{\pm} \Gamma\left(\frac{s+1\pm it_\phi}{2} \right) \Gamma\left(\frac{s+k-1\pm it_\phi}{2} \right) \Gamma\left(\frac{s+k\pm it_\phi}{2} \right) .\] Note that \(L(1/2,\operatorname{sym}^2 f \times \phi)\geq0\) (see [11]). We have the following approximate functional equations.

2.2 The Petersson trace formula↩︎

The Petersson trace formula is given by the following basic orthogonality relation on \(H_k\).

2.3 Fourier coefficients↩︎

Let \(\phi\) be a Hecke–Maass cusp form for \(\operatorname{SL}_2(\mathbb{Z})\) with the spectral parameter \(t_\phi\). We have the following strong bounds on the \(\operatorname{GL}(2)\) exponential sums.

2.4 An average of the \(J\)-Bessel function↩︎

We will use the following estimate of an average of the \(J\)-Bessel function.

By integrating by parts several times we get that \(\breve{W}(v)\ll (1+|v|)^{-B}\) and \(\hat{W}(v)\ll (1+|v|)^{-B}\) for any \(B \geq 0\).

3 The variance↩︎

In this section, we prove Theorem 1. For \(f\in H_k\) and \(g\in H_\ell\), by [6] we have \[\begin{gather} \frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})}\langle |F|^2,|G|^2\rangle - 1 \\ \ll \frac{1}{\sqrt{k\ell}} \sum_{t_\phi\ll \ell^{1/2+\varepsilon}} \frac{L(1/2,\phi)L(1/2,\operatorname{sym}^2f\times \phi)^{1/2}L(1/2,\operatorname{sym}^2g\times \phi)^{1/2}}{L(1,\operatorname{sym}^2f) L(1,\operatorname{sym}^2g) L(1,\operatorname{sym}^2 \phi)} \exp\left( - \frac{t_\phi^2}{2\ell}\right) \\ + \frac{1}{\sqrt{k\ell}} \int_{|t|\leq \ell^{1/2+\varepsilon}} \frac{|\zeta(1/2+it)|^2 |L(1/2+it,\operatorname{sym}^2 f)L(1/2+it,\operatorname{sym}^2 g)|}{L(1,\operatorname{sym}^2 f)L(1,\operatorname{sym}^2 g) |\zeta(1+2it)|^2 } \mathrm{d}t + k^{-2025}. \end{gather}\] Note that we have \(L(1/2,\phi)\geq0\) and \(L(1/2,\operatorname{sym}^2f\times \phi)\geq0\) (see [11], [15]). Hence by the Cauchy–Schwarz inequality, we have \[\label{eqn:var2S} \frac{1}{K^2} \sum_{K<k\leq 2K} \sum_{f\in H_k} \Big|\frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})}\langle |F_k|^2, |G_\ell|^2 \rangle-1 \Big|^2 \ll \mathcal{S}_c + \mathcal{S}_e + K^{-2025},\tag{6}\] where \[\label{eqn:Sc} \mathcal{S}_c := \frac{1}{K^3 \ell} \sum_{K<k\leq 2K} \sum_{f\in H_k} \Big|\sum_{t_\phi\ll \ell^{1/2+\varepsilon}} \frac{L(1/2,\phi)L(1/2,\operatorname{sym}^2f\times \phi)^{1/2}L(1/2,\operatorname{sym}^2g\times \phi)^{1/2}}{L(1,\operatorname{sym}^2f) L(1,\operatorname{sym}^2g) L(1,\operatorname{sym}^2 \phi)}\Big|^2\tag{7}\] and \[\label{eqn:Se} \mathcal{S}_e:= \frac{1}{K^3 \ell} \sum_{K<k\leq 2K} \sum_{f\in H_k} \Big| \int_{|t|\leq \ell^{1/2+\varepsilon}} \frac{|\zeta(1/2+it)|^2 |L(1/2+it,\operatorname{sym}^2 f)L(1/2+it,\operatorname{sym}^2 g)|}{L(1,\operatorname{sym}^2 f)L(1,\operatorname{sym}^2 g) |\zeta(1+2it)|^2 } \mathrm{d}t \Big|^2.\tag{8}\] We will estimate \(\mathcal{S}_c\) and \(\mathcal{S}_e\) separately.

3.1 The Eisentein series contribution↩︎

Note that by [16] and [17], we have \[\label{eqn:sym942-1} \zeta(1+2it) = (1+|t|)^{o(1)}, \quad L(1,\operatorname{sym}^2 f)=k^{o(1)}, \quad L(1,\operatorname{sym}^2 g) =\ell^{o(1)}.\tag{9}\] We have \[\mathcal{S}_e \ll \frac{K^\varepsilon}{K^3 \ell} \sum_{K<k\leq 2K} \sum_{f\in H_k} \Big| \int_{|t|\leq \ell^{1/2+\varepsilon}} |\zeta(1/2+it)|^2 |L(1/2+it,\operatorname{sym}^2 f)L(1/2+it,\operatorname{sym}^2 g)| \mathrm{d}t \Big|^2.\] By the Cauchy–Schwarz inequality, we have \[\begin{gather} \mathcal{S}_e \ll \frac{K^\varepsilon}{K^3 \ell} \sum_{K<k\leq 2K} \sum_{f\in H_k} \int_{|t_1|\leq \ell^{1/2+\varepsilon}} |\zeta(1/2+it_1)|^4 |L(1/2+it_1,\operatorname{sym}^2 f)|^2 \mathrm{d}t_1 \\ \cdot \int_{|t_2|\leq \ell^{1/2+\varepsilon}} |L(1/2+it_2,\operatorname{sym}^2 g)|^2 \mathrm{d}t_2 \\ = \frac{K^\varepsilon}{K^3 \ell} \int_{|t_1|\leq \ell^{1/2+\varepsilon}} |\zeta(1/2+it_1)|^4 \sum_{K<k\leq 2K} \sum_{f\in H_k} |L(1/2+it_1,\operatorname{sym}^2 f)|^2 \mathrm{d}t_1 \\ \cdot \int_{|t_2|\leq \ell^{1/2+\varepsilon}} |L(1/2+it_2,\operatorname{sym}^2 g)|^2 \mathrm{d}t_2. \end{gather}\]

Recall that we have the following well known bound \[\int_{|t|\leq T} |\zeta(1/2+it)|^4 \mathrm{d}t \ll T^{1+\varepsilon}.\] By Theorem 5 and Lemma 6, we have \[\label{eqn:Se6060} \mathcal{S}_e \ll \frac{K^\varepsilon}{K^3 \ell} K^2 \ell^{1/2+5/4} \ll \ell^{3/4} K^{\varepsilon-1}.\tag{10}\]

3.2 The cusp form contribution↩︎

By 7 , 9 , and the Cauchy–Schwarz inequality, we have \[\begin{gather} \mathcal{S}_c \ll \frac{K^\varepsilon}{K^3 \ell} \sum_{K<k\leq 2K} \sum_{f\in H_k} \sum_{t_\phi\ll \ell^{1/2+\varepsilon}} L(1/2,\phi) L(1/2,\operatorname{sym}^2f\times \phi) \\ \cdot \sum_{t_{\phi'} \ll \ell^{1/2+\varepsilon}} L(1/2,\phi') L(1/2,\operatorname{sym}^2g\times \phi') \exp\left( - \frac{t_{\phi'}^2}{\ell}\right)\\ = \frac{K^\varepsilon}{K^3 \ell} \sum_{t_\phi\ll \ell^{1/2+\varepsilon}} L(1/2,\phi) \left( \sum_{K<k\leq 2K} \sum_{f\in H_k} L(1/2,\operatorname{sym}^2f\times \phi) \right) \\ \cdot \sum_{t_{\phi'} \ll \ell^{1/2+\varepsilon}} L(1/2,\phi') L(1/2,\operatorname{sym}^2g\times \phi') \exp\left( - \frac{t_{\phi'}^2}{\ell}\right) . \end{gather}\]

Note that the spectral large sieve gives \[\sum_{t_\phi\ll \ell^{1/2+\varepsilon}} L(1/2,\phi) \ll \bigg( \sum_{t_\phi\ll \ell^{1/2+\varepsilon}} 1 \bigg)^{1/2} \bigg( \sum_{t_\phi\ll \ell^{1/2+\varepsilon}} L(1/2,\phi)^2 \bigg)^{1/2} \ll \ell^{1+\varepsilon}.\] Together with Theorems 4 and 6, we have \[\label{eqn:Sc6060} \mathcal{S}_c \ll \frac{K^\varepsilon}{K^3 \ell} \sum_{t_\phi\ll \ell^{1/2+\varepsilon}} L(1/2,\phi) K^{2+\varepsilon} \ell^{4/3+\varepsilon} \ll K^{-1+\varepsilon} \ell^{4/3.}\tag{11}\] Combining 6 , 10 and 11 , we complete the proof of Theorem 1.

3.3 An application to \(L\)-functions↩︎

Now we prove Corollary 2. It suffices to show that \[\label{eqn:60f2g262611} \frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})}\langle |F_k|^2, |G_\ell|^2 \rangle = 1 + O(K^{-\varepsilon}),\tag{12}\] for almost all even integer \(k\in(K,2K]\) and almost all \(f\in H_k\). Let \[\mathscr{S}_k := \sum_{f\in H_k} \Big|\frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})}\langle |F_k|^2, |G_\ell|^2 \rangle-1 \Big|^2.\] For \(\ell\leq K^{3/4-\varepsilon}\), Theorem 1 gives \(\sum_{K<k\leq2K} \mathscr{S}_k \ll K^{2-\varepsilon/3}.\) Hence we have \[\sum_{\substack{K<k\leq2K \\ \mathscr{S}_k\geq K^{1-\varepsilon/6}}} 1 \ll \sum_{\substack{K<k\leq2K }} \mathscr{S}_k / K^{1-\varepsilon/6} \ll K^{1-\varepsilon/6}.\] So for all but \(O(K^{1-\varepsilon/6})\) even integer \(k\in [K,2K]\), we have \(\mathscr{S}_k\leq K^{1-\varepsilon/6}\). For those \(k\), we have \[\sum_{\substack{f\in H_k \\ |\frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})}\langle |F_k|^2, |G_\ell|^2 \rangle-1| \geq K^{-\varepsilon/24}}} 1 \leq \sum_{\substack{f\in H_k}} \Big|\frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})}\langle |F_k|^2, |G_\ell|^2 \rangle-1 \Big|^2 /K^{-\varepsilon/12} \ll K^{1-\varepsilon/12}.\] Hence for all but \(O(K^{1-\varepsilon/12})\) forms \(f\in H_k\), we have \(|\frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})} \langle |F_k|^2, |G_\ell|^2 \rangle-1| \leq K^{-\varepsilon/24}\). This proves 12 , and hence Corollary 2 by using 3 and 9 .

4 Moments of \(L\)-functions↩︎

In this section, we prove Theorems 4, 5, and 6.

4.1 A first moment of the Rankin-Selberg \(L\)-functions↩︎

In this subsection, we will follow Luo–Sarnak’s method in [9] to prove Theorem 4. Let \(W\in C^\infty(\mathbb{R})\) such that \(\operatorname{supp}W\subset [1/2,3]\) and \(W^{(j)}(x)\ll_j 1\). It suffices to prove that for \(t_\phi\leq K^{1/2-\varepsilon}\), we have \[\label{eqn:M16060K} \mathcal{M}_1 := \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W\left(\frac{k-1}{K}\right) \frac{2\pi^2}{k-1}\sum_{f\in H_k} \frac{ L(1/2,\operatorname{sym}^2 f\times\phi)}{L(1,\operatorname{sym}^2 f)} \ll K^{1+\varepsilon}.\tag{13}\]

4.1.1 Applying the approximate functional equation↩︎

By Lemma 1, we have \[\mathcal{M}_1 = 2 \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W\left(\frac{k-1}{K}\right) \frac{2\pi^2}{k-1}\sum_{f\in H_k} \frac{1}{L(1,\operatorname{sym}^2 f)} \sum_{m,n\geq1} \frac{A(m,n)\lambda_\phi(n)}{(m^2n)^{1/2}} V_6(m^2n).\] By Lemma 2 and a smooth partition of unity, we arrive at \[\begin{gather} \label{eqn:M1} \mathcal{M}_1 = 2 \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W\left(\frac{k-1}{K}\right) \frac{2\pi^2}{k-1}\sum_{f\in H_k} \frac{1}{L(1,\operatorname{sym}^2 f)} \\ \cdot \sum_{m^2n \leq K^{2+\varepsilon}t_\phi} \frac{A(m,n)\lambda_\phi(n)}{(m^2n)^{1/2}} V_6(m^2n) + O(K^{-2025}) \\ \ll K^\varepsilon \sup_{s=\varepsilon+i\tau, \;\tau\in[-K^\varepsilon,K^\varepsilon]} \sup_{1\leq N \leq K^{2+\varepsilon}t_\phi} | \mathcal{M}_1(s,N) | + 1, \end{gather}\tag{14}\] where \[\mathcal{M}_1(s,N) := \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) \frac{2\pi^2}{k-1}\sum_{f\in H_k} \frac{1}{L(1,\operatorname{sym}^2 f)} \sum_{m^2n \geq1 } \frac{A(m,n)\lambda_\phi(n)}{(m^2n)^{1/2}} V_s\left(\frac{m^2n}{N}\right).\] Here \(W_s \in \mathcal{C}_c^\infty(\mathbb{R})\) with \[\label{eqn:WVs} \operatorname{supp}W_s\subset [1/2,3] \quad \textrm{ and } \quad W_s^{(j)}(x) \ll_j K^{j\varepsilon}.\tag{15}\] We have exactly the same properties for \(V_s\). By 4 and rearranging the sums, we get \[\begin{gather} \mathcal{M}_1(s,N) = \sum_{m^2n \geq 1} \frac{\lambda_\phi(n)}{(m^2n)^{1/2 }} V_s\left(\frac{m^2n}{N}\right) \sum_{d\mid (m,n)} \mu(d) \\ \cdot \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) \frac{2\pi^2}{k-1}\sum_{f\in H_k} \frac{1}{L(1,\operatorname{sym}^2 f)} A(m/d,1) A(1,n/d). \end{gather}\] Making changes of variables, we get \[\begin{gather} \mathcal{M}_1(s,N) = \sum_{d^3 m^2n \geq 1} \frac{\lambda_\phi(dn)}{(d^3m^2n)^{1/2 }} \mu(d) V_s\left(\frac{d^3m^2n}{N}\right) \sum_{m_1^2 m_2 =m} \sum_{n_1^2 n_2 =n} \\ \cdot \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) \frac{2\pi^2}{k-1}\sum_{f\in H_k} \frac{\lambda_f(m_2^2) \lambda_f(n_2^2)}{L(1,\operatorname{sym}^2 f)} \\ = \sum_{d^3 m_1^4 m_2^2 n_1^2 n_2 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2)}{(d^3m_1^4 m_2^2 n_1^2 n_2)^{1/2 }} \mu(d) V_s\left(\frac{d^3 m_1^4 m_2^2 n_1^2 n_2}{N}\right) \\ \cdot \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) \frac{2\pi^2}{k-1}\sum_{f\in H_k} \frac{\lambda_f(m_2^2) \lambda_f(n_2^2)}{L(1,\operatorname{sym}^2 f)} . \end{gather}\]

4.1.2 Applying the Petersson trace formula↩︎

By Lemma 3, we get \[\label{eqn:M161M1043M11} \mathcal{M}_1(s,N) = \mathcal{M}_{10}(s,N) + \mathcal{M}_{11}(s,N) ,\tag{16}\] where the diagonal contribution is \[\mathcal{M}_{10}(s,N) := \sum_{d^3 m_1^4 m_2^2 n_1^2 n_2 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2) \delta_{m_2,n_2} }{(d^3m_1^4 m_2^2 n_1^2 n_2)^{1/2 }} \mu(d) V_s\left(\frac{d^3 m_1^4 m_2^2 n_1^2 n_2}{N}\right) \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) ,\] and the terms involving the \(J\)-Bessel function is \[\begin{gather} \mathcal{M}_{11}(s,N) := \sum_{d^3 m_1^4 m_2^2 n_1^2 n_2 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2)}{(d^3m_1^4 m_2^2 n_1^2 n_2)^{1/2+s}} \mu(d) V_s\left(\frac{d^3 m_1^4 m_2^2 n_1^2 n_2}{N}\right) \\ \cdot \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) 2\pi i^{-k} \sum_{c=1}^{\infty} \frac{S(m_2^2,n_2^2;c)}{c}J_{k-1}\left(\frac{4\pi m_2n_2}{c}\right) . \end{gather}\]

4.1.3 The diagonal contribution↩︎

We first deal with \(\mathcal{M}_{10}(s,N)\). We have \[\mathcal{M}_{10}(s,N) = \sum_{d^3 m_1^4 n_1^2 n_2^3 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2)}{(d^3m_1^4 n_1^2 n_2^3)^{1/2}} \mu(d) V_s\left(\frac{d^3 m_1^4 n_1^2 n_2^3}{N}\right) \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) .\] By the Mellin inversion formula, we get \[\mathcal{M}_{10}(s,N) = \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) \frac{1}{2\pi i} \int_{(2)} \sum_{d^3 m_1^4 n_1^2 n_2^3 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2) \mu(d) }{(d^3m_1^4 n_1^2 n_2^3)^{1/2+w}} \tilde{V_s}(w) N^w \mathrm{d}w .\] Note that \[\sum_{d^3 m_1^4 n_1^2 n_2^3 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2) \mu(d) }{(d^3m_1^4 n_1^2 n_2^3)^{1/2+w}} = \sum_{n \geq 1} \Big( \sum_{d^3 m_1^4 n_1^2 n_2^3 =n} \lambda_\phi(dn_1^2 n_2) \mu(d) \Big) n^{-1/2-w}.\] Writing \(dn_2=m\), we get \[\sum_{d^3 m_1^4 n_1^2 n_2^3 =n} \lambda_\phi(dn_1^2 n_2) \mu(d) = \sum_{m^3 m_1^4 n_1^2 =n} \lambda_\phi(m n_1^2 ) \sum_{d\mid m}\mu(d) = \sum_{ m_1^4 n_1^2 =n} \lambda_\phi( n_1^2 ).\] For \(\operatorname{Re}(w)>2\) we have \[\sum_{d^3 m_1^4 n_1^2 n_2^3 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2) \mu(d) }{(d^3m_1^4 n_1^2 n_2^3)^{1/2+w}} = L(1+2w,\operatorname{sym}^2\phi).\] Hence we get \[\label{eqn:M106060} \mathcal{M}_{10}(s,N) = \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) \frac{1}{2\pi i} \int_{(\varepsilon)} L(1+2w,\operatorname{sym}^2\phi) \tilde{V_s}(w) N^w \mathrm{d}w \ll K^{1+\varepsilon} .\tag{17}\]

4.1.4 The Bessel function contribution↩︎

Now we treat \(\mathcal{M}_{11}(s,N)\). Rearranging the order of the sums, we get \[\begin{gather} \mathcal{M}_{11}(s,N) = \pi \sum_{d^3 m_1^4 m_2^2 n_1^2 n_2 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2)}{(d^3m_1^4 m_2^2 n_1^2 n_2)^{1/2+s}} \mu(d) V_s\left(\frac{d^3 m_1^4 m_2^2 n_1^2 n_2}{N}\right) \\ \cdot \sum_{c=1}^{\infty} \frac{S(m_2^2,n_2^2;c)}{c} \sum_{k\equiv 0 \; \mathrm{mod}\; 2} W_s\left(\frac{k-1}{K}\right) 2 i^{-k} J_{k-1}\left(\frac{4\pi m_2n_2}{c}\right) . \end{gather}\] By Lemma 5, we have \[\label{eqn:M1160M11143M112} \mathcal{M}_{11}(s,N) \ll |\mathcal{M}_{111}(s,N)|+|\mathcal{M}_{112}(s,N)| + \mathcal{M}_{113}(N) ,\tag{18}\] where \[\begin{gather} \mathcal{M}_{111}(s,N) := \sum_{d^3 m_1^4 m_2^2 n_1^2 n_2 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2)}{(d^3m_1^4 m_2^2 n_1^2 n_2)^{1/2+s}} \mu(d) V_s\left(\frac{d^3 m_1^4 m_2^2 n_1^2 n_2}{N}\right) \\ \cdot \sum_{c=1}^{\infty} \frac{S(m_2^2,n_2^2;c)}{c^{1/2}} \frac{K}{(m_2n_2)^{1/2}} e\left(\frac{2 m_2n_2}{c}\right) \breve{W}_s\left(\frac{K^2 c}{8\pi m_2n_2}\right) , \end{gather}\] \[\begin{gather} \mathcal{M}_{112}(s,N) := \sum_{d^3 m_1^4 m_2^2 n_1^2 n_2 \geq 1} \frac{\lambda_\phi(dn_1^2 n_2)}{(d^3m_1^4 m_2^2 n_1^2 n_2)^{1/2+s}} \mu(d) V_s\left(\frac{d^3 m_1^4 m_2^2 n_1^2 n_2}{N}\right) \\ \cdot \sum_{c=1}^{\infty} \frac{S(m_2^2,n_2^2;c)}{c^{1/2}} \frac{K}{(m_2n_2)^{1/2}} e\left(-\frac{2 m_2n_2}{c}\right) \overline{\breve{W}_s\left(\frac{K^2 c}{8\pi m_2n_2}\right) } , \end{gather}\] and \[\mathcal{M}_{113}(N) := K^\varepsilon \sum_{d^3 m_1^4 m_2^2 n_1^2 n_2 \asymp N} \frac{|\lambda_\phi(dn_1^2 n_2)|}{(d^3m_1^4 m_2^2 n_1^2 n_2)^{1/2}} \sum_{c=1}^{\infty} \frac{|S(m_2^2,n_2^2;c)|}{c} \frac{1}{K^4} \frac{m_2n_2}{c} .\]

We first deal with \(\mathcal{M}_{113}(N)\). By Weil’s bound on the Kloosterman sums, we have \[\mathcal{M}_{113}(N) \ll K^{-4+\varepsilon} \sum_{d^3 m_1^4 m_2^2 n_1^2 n_2 \asymp N} \frac{|\lambda_\phi(dn_1^2 n_2)|}{(d^3m_1^4 n_1^2 )^{1/2}} n_2^{1/2} \sum_{c=1}^{\infty} \frac{ (m_2^2,n_2^2,c)^{1/2} }{c^{3/2-\varepsilon}} .\] Using the bound \((m_2^2,n_2^2,c)\leq m_2^2\) and Kim–Sarnak’s bounds \(|\lambda_\phi(n)|\ll n^{7/64+\varepsilon}\), we get \[\begin{gather} \label{eqn:M1136060} \mathcal{M}_{113}(N) \ll K^{-4+\varepsilon} \sum_{d^3 m_1^4 m_2^2 n_1^2 n_2 \asymp N} \frac{ (dn_1^2 n_2)^{7/64+\varepsilon}}{(d^3m_1^4 n_1^2 )^{1/2}} m_2 n_2^{1/2} \sum_{c=1}^{\infty} \frac{ 1}{c^{3/2-\varepsilon}} \\ \ll N^{2-\varepsilon} K^{-4+\varepsilon} \ll t_\phi^2 \ll K , \end{gather}\tag{19}\] provided by \(N \leq K^{2+\varepsilon}t_\phi\) and \(t_\phi\leq K^{1/2}\).

We now treat \(\mathcal{M}_{111}(s,N)\). The estimate for \(\mathcal{M}_{112}(s,N)\) will be the same. We have \[\begin{gather} \mathcal{M}_{111}(s,N) = N^{-s} K^{-1} \sum_{c=1}^{\infty} \frac{1}{c^{3/2}} \sum_{d^3 m_1^4 m_2^2 n_1^2 \geq 1} \mu(d) \frac{1}{(d^3m_1^4 m_2 n_1^2 )^{1/2}} \\ \cdot \sum_{n_2\geq1} \lambda_\phi(dn_1^2 n_2) S(m_2^2,n_2^2;c) e\left(\frac{2 m_2n_2}{c}\right) \mathcal{V}\left(\frac{d^3 m_1^4 m_2^2 n_1^2 n_2}{N}\right) \mathcal{W}\left(\frac{K^2 c}{ m_2n_2}\right) , \end{gather}\] where \(\mathcal{V}(y)=y^{-s}V_s(y)\) and \(W_1(y)= y \breve{W}_s(y/8\pi)\). Note that by 15 we have \[\label{eqn:cV} \operatorname{supp}\mathcal{V}\subset [1/2,3] \quad \textrm{ and } \quad \mathcal{V}^{(j)}(y) \ll_j K^{j\varepsilon }.\tag{20}\] By the definition of \(\breve{W}_s\) and repeated integration by parts, we know \[\breve{W}_s^{(j)}(y) \ll_{j,A} \left(\frac{K^\varepsilon}{1+|y|}\right)^{A}, \quad \textrm{for any A\geq0.}\] Hence we have \[\label{eqn:cW} \mathcal{W}^{(j)}(y) \ll_{j,A} K^\varepsilon \left(\frac{K^\varepsilon}{1+|y|}\right)^{A}, \quad \textrm{for any A\geq0.}\tag{21}\] These show that \(m_2n_2\leq 3N\) and the contribution from \(K^2c/(m_2n_2)\leq K^{2\varepsilon}\) is negligibly small. So we can truncate the \(c\)-sum at \(c\ll N K^{\varepsilon-2}\).

By the Hecke relations, we get \(\lambda_\phi(dn_1^2 n_2) =\sum_{a\mid (dn_1^2,n_2)}\mu(a)\lambda_\phi(dn_1^2/a) \lambda_\phi(n_2/a)\). Writing \(n_2=an\), we obtain \[\begin{gather} \mathcal{M}_{111}(s,N) = N^{-s} K^{-1} \sum_{c\ll N K^{\varepsilon-2}} \frac{1}{c^{3/2}} \sum_{d^3 m_1^4 m_2^2 n_1^2 \geq 1} \mu(d) \frac{1}{(d^3m_1^4 m_2 n_1^2 )^{1/2}} \sum_{a\mid dn_1^2}\mu(a)\lambda_\phi(dn_1^2/a) \\ \cdot \sum_{n\geq1} \lambda_\phi(n) S(m_2^2,a^2 n^2;c) e\left(\frac{2 m_2an}{c}\right) \mathcal{V}\left(\frac{d^3 m_1^4 m_2^2 n_1^2 an}{N}\right) \mathcal{W}\left(\frac{K^2 c}{ m_2an}\right) + O(K^{-B}) , \end{gather}\] Breaking the \(n\)-sum into arithmetic progressions modulo \(c\), we get \[\begin{gather} \mathcal{M}_{111}(s,N) = N^{-s} K^{-1} \sum_{c\ll N K^{\varepsilon-2}} \frac{1}{c^{3/2}} \sum_{d^3 m_1^4 m_2^2 n_1^2 \geq 1} \mu(d) \frac{1}{(d^3m_1^4 m_2 n_1^2 )^{1/2}} \\ \cdot \sum_{a\mid dn_1^2}\mu(a)\lambda_\phi(dn_1^2/a) \sum_{\alpha\operatorname{mod}\; c} S(m_2^2,a^2 \alpha^2;c) e\left(\frac{2 m_2a \alpha}{c}\right) \\ \cdot \sum_{\substack{n\geq1 \\ n\equiv \alpha \operatorname{mod}\; c}} \lambda_\phi(n) \mathcal{V}\left(\frac{d^3 m_1^4 m_2^2 n_1^2 an}{N}\right) \mathcal{W}\left(\frac{K^2 c}{ m_2an}\right) + O(K^{-B}) , \end{gather}\] By using the additive characters modulo \(c\), we know that the innermost \(n\)-sum above is \[\begin{align} S_1 & = \sum_{\substack{n\geq1 \\ n\equiv \alpha \operatorname{mod}\; c}} \lambda_\phi(n) \mathcal{V}\left(\frac{d^3 m_1^4 m_2^2 n_1^2 an}{N}\right) \mathcal{W}\left(\frac{K^2 c}{ m_2an}\right) \\ & = \frac{1}{c} \sum_{\beta\operatorname{mod}\; c} e( - \alpha \beta/c) \sum_{ n\geq1 } \lambda_\phi(n) e(n\beta/c) \mathcal{V}\left(\frac{d^3 m_1^4 m_2^2 n_1^2 an}{N}\right) \mathcal{W}\left(\frac{K^2 c}{ m_2an}\right) . \end{align}\] By the partial summation formula we get \[S_1 \leq \max_{\beta\operatorname{mod}\; c} \bigg| \int_{\frac{N}{4d^3 m_1^4 m_2^2 n_1^2 a}} ^{\frac{4N}{d^3 m_1^4 m_2^2 n_1^2 a}} \bigg(\sum_{ n\leq u } \lambda_\phi(n) e(n\beta/c)\bigg) \bigg(\mathcal{V}\left(\frac{d^3 m_1^4 m_2^2 n_1^2 a u}{N}\right) \mathcal{W}\left(\frac{K^2 c}{ m_2a u}\right) \bigg)' \mathrm{d}u \bigg|.\] By Lemma 4 we have \[S_1 \ll \int_{\frac{N}{4d^3 m_1^4 m_2^2 n_1^2 a}} ^{\frac{4N}{d^3 m_1^4 m_2^2 n_1^2 a}} u^{1/2} t_\phi^{1/2+\varepsilon} \bigg(\mathcal{V}\left(\frac{d^3 m_1^4 m_2^2 n_1^2 a u}{N}\right) \mathcal{W}\left(\frac{K^2 c}{ m_2a u}\right) \bigg)' \mathrm{d}u .\] By 20 and 21 we get \[S_1 \ll K^\varepsilon \left(\frac{N}{d^3 m_1^4 m_2^2 n_1^2 a}\right)^{1/2 } t_\phi^{1/2}.\] Hence \[\begin{gather} \mathcal{M}_{111}(s,N) \ll K^{-1+\varepsilon} \sum_{c\ll N K^{\varepsilon-2}} \frac{1}{c^{3/2}} \sum_{d^3 m_1^4 m_2^2 n_1^2 \ll N} \frac{1}{(d^3m_1^4 m_2 n_1^2 )^{1/2}} \sum_{a\mid dn_1^2} \left(\frac{dn_1^2}{a}\right)^{1/2} \\ \cdot \sum_{\alpha\operatorname{mod}\; c} |S(m_2^2,a^2 \alpha^2;c)| \left(\frac{N}{d^3 m_1^4 m_2^2 n_1^2 a}\right)^{1/2 } t_\phi^{1/2} . \end{gather}\] By Weil’s bound on the Kloosterman sums, we get \[\begin{align} \mathcal{M}_{111}(s,N) & \ll \frac{ N^{1/2} t_\phi^{1/2} }{K^{1-\varepsilon}} \sum_{c\ll N K^{\varepsilon-2}} \sum_{ m_2 \ll N^{1/2}} \frac{1}{m_2^{3/2}} (m_2^2,c)^{1/2} \nonumber \\ & \leq \frac{ N^{1/2} t_\phi^{1/2} }{K^{1-\varepsilon}} \sum_{ m_2 \ll N^{1/2}} \frac{1}{m_2^{3/2}} \sum_{c\ll N K^{\varepsilon-2}} \sum_{d\mid (m_2^2,c)} d^{1/2} \nonumber \\ & \leq \frac{ N^{1/2} t_\phi^{1/2} }{K^{1-\varepsilon}} \sum_{ m_2 \ll N^{1/2}} \frac{1}{m_2^{3/2}} \sum_{d\mid m_2^2} d^{-1/2} N K^{\varepsilon-2} \ll \frac{ N^{3/2}t_\phi^{1/2}}{ K^{3-\varepsilon}} \ll K^\varepsilon t_\phi^2. \label{eqn:M1116060} \end{align}\tag{22}\] In the last inequality, we have used the condition \(N \leq K^{2+\varepsilon}t_\phi\).

By 14 , 16 , 17 , 18 , 19 , 22 , we prove 13 . Hence we complete the proof of Theorem 4.

4.2 A second moment of the symmetric square \(L\)-functions↩︎

In this subsection, we prove Theorem 5. Cf. Khan [14]. Let \(W\in C^\infty(\mathbb{R})\) such that \(\operatorname{supp}W\subset [1/2,3]\) and \(W^{(j)}(x)\ll_j 1\). It suffices to prove that for \(-K^{1/2-\varepsilon}\leq t\leq K^{1/2-\varepsilon}\), we have \[\mathcal{M}_2 := \sum_{k\geq12} W\left(\frac{k-1}{K}\right) \sum_{f\in H_k} \frac{| L(1/2+it,\operatorname{sym}^2 f)|^2 }{L(1,\operatorname{sym}^2 f)} \ll K^{2+\varepsilon},\] for any \(\varepsilon>0\).

4.2.1 Applying the approximate functional equation↩︎

By Lemmas 1 and 2, and a smooth partition of unity, we get \[\mathcal{M}_2 \ll \sup_{N\leq K^{1+\varepsilon}\sqrt{T}} \sum_{k\geq12} W\left(\frac{k-1}{K}\right) \sum_{f\in H_k} \frac{1 }{L(1,\operatorname{sym}^2 f)} \left| \sum_{n\geq1} \frac{\lambda_f(n^2)}{n^{1/2+it}} V_3^+\left(\frac{n}{N}\right) V_1\left(\frac{n}{N}\right) \right|^2 + 1,\] where \(V_1(\xi)\in \mathcal{C}_c^\infty(\mathbb{R})\) with \(\operatorname{supp}V_1\subseteq [1,2]\), \(V_1^{(j)}(\xi)\ll_j 1\), for any \(j\in \mathbb{Z}_{\geq0}\). Here we write \(T=1+|t|\). Hence by Lemma 2 again, we obtain \[\begin{gather} \mathcal{M}_2 \ll K^{\varepsilon} \sup_{N\leq K^{1+\varepsilon}\sqrt{T}} \sideset{}{^{\rm even}} \sum_{k\geq12} W\left(\frac{k-1}{K}\right) \sum_{f\in H_k} \frac{1 }{L(1,\operatorname{sym}^2 f)} \\ \cdot \int_{\varepsilon-i K^{\varepsilon}}^{\varepsilon+i K^{\varepsilon}} \left| \sum_{n\geq1} \frac{\lambda_f(n^2)}{n^{1/2+it}} \left(\frac{n}{N}\right)^{-s} V_1\left(\frac{n}{N}\right) \right|^2 \mathrm{d}s + 1. \end{gather}\] Hence we have \[\label{eqn:M26060} \mathcal{M}_2 \ll K^{1+\varepsilon} \sup_{N\leq K^{1+\varepsilon}\sqrt{T}} \mathcal{M}_2(N) + 1,\tag{23}\] where \[\mathcal{M}_2(N):= \sideset{}{^{\rm even}} \sum_{k\geq12} W\left(\frac{k-1}{K}\right) \frac{12\zeta(2)}{(k-1)}\sum_{f\in H_k} \frac{1 }{L(1,\operatorname{sym}^2 f)} \bigg| \sum_{n\geq1} \frac{\lambda_f(n^2)}{n^{1/2+it}} V\left(\frac{n}{N}\right) \bigg|^2,\] for certain \(V(\xi)\in \mathcal{C}_c^\infty(\mathbb{R})\) with \(\operatorname{supp}V\subseteq [1,2]\), \(V^{(j)}(\xi)\ll_j K^{j\varepsilon}\), for any \(j\in \mathbb{Z}_{\geq0}\). Opening the square and rearranging the sums, we have \[\begin{gather} \mathcal{M}_2(N)= \sum_{m\geq1} \frac{1}{m^{1/2+it}} V\left(\frac{m}{N}\right) \sum_{n\geq1} \frac{1}{n^{1/2-it}} \overline{V\left(\frac{n}{N}\right)} \\ \cdot \sideset{}{^{\rm even}}\sum_{k\geq12} W\left(\frac{k-1}{K}\right) \frac{12\zeta(2)}{(k-1)}\sum_{f\in H_k} \frac{\lambda_f(m^2)\lambda_f(n^2)}{L(1,\operatorname{sym}^2 f)}. \end{gather}\]

4.2.2 Applying the Petersson trace formula↩︎

By Lemma 3, the second line of the above equation is equal to \[\sideset{}{^{\rm even}}\sum_{k\geq12} W\left(\frac{k-1}{K}\right) \delta_{m,n} + \sideset{}{^{\rm even}}\sum_{k\geq12} W\left(\frac{k-1}{K}\right) 2\pi i^{-k} \sum_{c=1}^{\infty} \frac{S(m^2,n^2;c)}{c}J_{k-1}\left(\frac{4\pi mn}{c}\right) .\] Hence we have \[\label{eqn:M261M2043M21} \mathcal{M}_2(N)=\mathcal{M}_{20}(N)+\mathcal{M}_{21}(N),\tag{24}\] where the diagonal contribution is \[\mathcal{M}_{20}(N) := \sum_{n\geq1} \frac{1}{n} \left|V\left(\frac{n}{N}\right)\right|^2 \sideset{}{^{\rm even}}\sum_{k\geq12} W\left(\frac{k-1}{K}\right) \ll K.\] and the terms involving the \(J\)-Bessel function is \[\begin{gather} \mathcal{M}_{21}(N) := \pi \sum_{m\geq1} \frac{1}{m^{1/2+it}} V\left(\frac{m}{N}\right) \sum_{n\geq1} \frac{1}{n^{1/2-it}} \overline{V\left(\frac{n}{N}\right)} \\ \cdot \sum_{c=1}^{\infty} \frac{S(m^2,n^2;c)}{c} \sideset{}{^{\rm even}}\sum_{k\geq12} W\left(\frac{k-1}{K}\right) 2 i^{-k} J_{k-1}\left(\frac{4\pi mn}{c}\right). \end{gather}\]

4.2.3 The Bessel function contribution↩︎

By Lemma 5, we have \[\begin{gather} \mathcal{M}_{21}(N) = \pi \sum_{m\geq1} \frac{1}{m^{1/2+it}} V\left(\frac{m}{N}\right) \sum_{n\geq1} \frac{1}{n^{1/2-it}} \overline{V\left(\frac{n}{N}\right)} \sum_{c=1}^{\infty} \frac{S(m^2,n^2;c)}{c} \\ \cdot \left(-\frac{K}{\sqrt{x}} \operatorname{Im}\left\{ e(-1/8) e^{ix} \breve{W}(K^2/2x) \right\} + O\left(\frac{x}{K^4} \int_{\mathbb{R}} v^4 |\hat{W}(v)| \mathrm{d}v \right) \right), \end{gather}\] where \(x=4\pi mn/c\). The contribution from the error term is bounded by \[O\left( \sum_{m\ll N} \frac{1}{m^{1/2}} \sum_{n\ll N} \frac{1}{n^{1/2}} \sum_{c=1}^{\infty} \frac{(m^2,n^2,c)^{1/2} c^{1/2+\varepsilon}}{c} \frac{mn}{cK^4} \right) = O\left( \frac{N^2}{K^4} \sum_{n\ll N} \sum_{c=1}^{\infty} \frac{(n^2,c)^{1/2}}{c^{3/2-\varepsilon}} \right).\] Note that \(\sum_{n\ll N} \sum_{c=1}^{\infty} \frac{(n^2,c)^{1/2}}{c^{3/2-\varepsilon} } \ll \sum_{n\ll N} \sum_{d\mid n^2} \sum_{\substack{c\geq1,d\mid c}} \frac{d^{1/2}}{c^{3/2-\varepsilon}} \ll N\). The above is \[O(N^3 K^{-4}) = O(T^{3/2}K^{-1+\varepsilon}),\] which is \(O(K)\) if \(T\leq K\). Hence we get \[\label{eqn:M26060M211} \mathcal{M}_2(N) \ll \mathcal{M}_{211}(N) + K,\tag{25}\] where \(\mathcal{M}_{211}(N)\) is defined by \[K \sum_{c=1}^{\infty} \frac{1}{c^{1/2}} \sum_{m\geq1} \frac{1}{m^{1+it}} V\left(\frac{m}{N}\right) \sum_{n\geq1} \frac{1}{n^{1-it}} \overline{V\left(\frac{n}{N}\right)} S(m^2,n^2;c) e\left(\pm \frac{2mn}{c}\right) \breve{W}\left(\frac{K^2c}{8\pi mn}\right) .\] Note that by \(\breve{W}(v)\ll (1+|v|)^B\), the contribution from terms with \(N^2/c \leq K^{2-\varepsilon}\) is negligibly small. Hence we can truncate the \(c\)-sum at \(c\leq N^2 /K^{2-\varepsilon}\), getting \[\begin{gather} \label{eqn:M2116060} \mathcal{M}_{211}(N) = K \sum_{c\leq N^2 /K^{2-\varepsilon}} \frac{1}{c^{1/2}} \sum_{m\geq1} \frac{1}{m^{1+it}} V\left(\frac{m}{N}\right) \\ \cdot\sum_{n\geq1} \frac{1}{n^{1-it}} \overline{V\left(\frac{n}{N}\right)} S(m^2,n^2;c) e\left(\pm \frac{2mn}{c}\right) \breve{W}\left(\frac{K^2c}{8\pi mn}\right) + O_B(K^{-B}), \end{gather}\tag{26}\] for any \(B>0\).

If \(|t|\leq K^\varepsilon\), then we have \[\mathcal{M}_{211}(N) \ll K \sum_{c\ll K^\varepsilon} c \sum_{m\asymp N}\frac{1}{m}\sum_{n\asymp N}\frac{1}{n} \ll K^{1+\varepsilon}.\]

If \(K^\varepsilon \leq |t| \leq K^{1/2-\varepsilon}\), then we consider the \(n\)-sum in \(\mathcal{M}_{211}(N)\), \[\begin{align} \mathcal{S} & = \sum_{n\geq1} S(m^2,n^2;c) e\left(\pm \frac{2mn}{c}\right) \frac{1}{n^{1-it}} \overline{V\left(\frac{n}{N}\right)} \breve{W}\left(\frac{K^2c}{8\pi mn}\right) \\ & = \sum_{b\operatorname{mod}\; c} S(m^2,b^2;c) e\left(\pm \frac{2mb}{c}\right) \sum_{\substack{n\geq1\\n\equiv b\operatorname{mod}\; c}} \frac{1}{n^{1-it}} \overline{V\left(\frac{n}{N}\right)} \breve{W}\left(\frac{K^2c}{8\pi mn}\right). \end{align}\] By the Poisson summation formula we get \[\mathcal{S} = \sum_{b\operatorname{mod}\; c} S(m^2,b^2;c) e\left(\pm \frac{2mb}{c}\right) \frac{1}{c} \sum_{n\in \mathbb{Z}} e\left(\frac{nb}{c}\right) \mathcal{I}(n),\] where \[\mathcal{I}(n) := \int_{\mathbb{R}} \frac{1}{y^{1-it}} \overline{V\left(\frac{y}{N}\right)} \breve{W}\left(\frac{K^2c}{8\pi my}\right) e\left(-\frac{ny}{c}\right) \mathrm{d}y.\] By making a change of variable \(y=N\xi\), we have \[\mathcal{I}(n) = N^{it} \int_{\mathbb{R}} \frac{1}{\xi} \overline{V\left(\xi\right)} \breve{W}\left(\frac{K^2c}{8\pi mN\xi}\right) e\left(\frac{t}{2\pi}\log \xi - \frac{nN}{c}\xi\right) \mathrm{d}\xi.\] By repeated integration by parts and the assumption \(|t|\leq K^\varepsilon\), we have \[\mathcal{I}(0) \ll_B K^{-B},\] for any \(B>0\). Recall that \(N\leq K^{1+\varepsilon}\sqrt{T}\). For \(|n|\geq1\) and \(c\leq N^2/K^{2-\varepsilon}\), we have \(|nN/c|\geq N/c\geq K^{2-\varepsilon}/N \geq K^{1-2\varepsilon}/\sqrt{T}\). If \(T\ll K^{1/2}\), we have \(|nN/c|\gg K^\varepsilon T\). By repeated integration by parts, we have \[\mathcal{I}(n) \ll_B n^{-6} K^{-B},\] for any \(B>0\). Hence by 26 we have \(\mathcal{M}_{211}(N) \ll_B K^{-B}\), for any \(B>0\).

Combining 23 , 24 , and 25 , we complete the proof of Theorem 5.

4.3 A mixed moment of \(L\)-functions↩︎

In this subsection, we prove Theorem 6. By [6] we know that for \(|t|\leq \ell^{2/3}\), \[\frac{\Gamma(\ell-1/2+it)}{\Gamma(\ell)} \asymp \frac{1}{\ell^{1/2}} \exp\left(-\frac{t^2}{2\ell}\right).\] By the standard Rankin–Selberg method and the Watson formula [18], we have \[\begin{gather} \|g\|_4^4 \asymp 1 + \frac{1}{\ell} \;\;\; \sideset{}{^{\rm even}}\sum_{t_\phi\leq \ell^{1/2+\varepsilon}} \frac{L(1/2,\phi) L(1/2,\operatorname{sym}^2 g\times \phi)}{L(1,\operatorname{sym}^2 g)^2 L(1,\operatorname{sym}^2 \phi)} \exp\left(-\frac{t_\phi^2}{\ell}\right) \\ + \frac{1}{\ell} \int_{|t|\leq \ell^{1/2+\varepsilon}} \frac{|\zeta(1/2+it)|^2 |L(1/2+it,\operatorname{sym}^2 g)|^2}{ L(1,\operatorname{sym}^2 g)^2 |\zeta(1+2it)|^2 } \exp\left(-\frac{t^2}{\ell}\right) \mathrm{d}t . \end{gather}\] Hence by Blomer–Khan–Young’s \(L^4\)-norm bound 1 we have \[\sideset{}{^{\rm even}} \sum_{t_\phi\leq \ell^{1/2+\varepsilon}} \frac{L(1/2,\phi) L(1/2,\operatorname{sym}^2 g\times \phi)}{L(1,\operatorname{sym}^2 g)^2 L(1,\operatorname{sym}^2 \phi)} \exp\left(-\frac{t_\phi^2}{\ell}\right) \\ \ll \ell \|g\|_4^4 \ll \ell^{4/3+\varepsilon} .\] By 9 , we complete the proof of Theorem 6.

Acknowledgements↩︎

The author wants to thank Professors Jianya Liu and Zeév Rudnick for encouragements.

References↩︎