On the quantitative simultaneous sign changes of Fourier coefficients attached to Hecke–Maass forms over certain quadratic forms

Guodong Hua

doi:10.4153/S0008439526102185

On the quantitative simultaneous sign changes of Fourier coefficients attached to Hecke–Maass forms over certain quadratic forms

Part of: Multiplicative number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press: 26 June 2026

Guodong Hua

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Guodong Hua*: Affiliation:
Weinan Normal University, China
*: e-mail: gdhuasdu@163.com

Article contents

Abstract
Introduction
Preliminaries
The main propositions
Proof of Theorem
Data availability statement
Competing interests
Footnotes
References

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Abstract

Let f and g be two distinct Hecke–Maass cusp forms of weight zero for the full modular group $SL(2,\mathbb {Z})$ with Laplacian eigenvalues $\frac {1}{4}+t_{1}^{2}$ and $\frac {1}{4}+t_{2}^{2}$, and denote by $\lambda _{f}(n)$ and $\lambda _{g}(n)$ the n-th normalized Fourier coefficients attached to f and g, respectively. In the present article, we establish a quantitative result regarding the sign changes of the sequence $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n\geqslant 1}$ in the interval $(x,2x]$, supported at a certain primitive integral binary quadratic form with fixed negative discriminant $D<0$, for sufficiently large x.

Keywords

Hecke–Maass form Fourier coefficients sign changes binary quadratic forms

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces

Secondary: 11N37: Asymptotic results on arithmetic functions

Information

Type: Article
Information: Canadian Mathematical Bulletin , First View , pp. 1 - 20

DOI: https://doi.org/10.4153/S0008439526102185 [Opens in a new window]
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

1 Introduction

The oscillation of Fourier coefficients of automorphic forms is an intriguing and significant topic in modern number theory, which has been investigated extensively in the literature (see, e.g., [Reference Acharya1, Reference Babita and Vaishya2, Reference Hecke10, Reference Hua14, Reference Vaishya54]). Let $H_{k}^{*}$ be the set of all normalized primitive Hecke cusp forms of even integral weight k for the full modular group $\Gamma =SL(2,\mathbb {Z})$ . Then, given any normalized cuspidal Hecke eigenform $\phi \in H_{k}^{*}$ , it admits the Fourier expansion at the cusp $\infty $ :

$$ \begin{align*} \phi(\tau) = \sum_{n=1}^{\infty}\lambda_{\phi}(n)n^{\frac{k-1}{2}}q^{n}, \qquad q:=e^{2\pi i\tau}, \end{align*} $$ StartLayout 1st Row phi left parenthesis tau right parenthesis equals sigma summation Underscript n equals 1 Overscript infinity Endscripts lamda Subscript phi Baseline left parenthesis n right parenthesis n Superscript StartFraction k minus 1 Over 2 EndFraction Baseline q Superscript n Baseline comma q colon equals e Superscript 2 pi i tau Baseline comma EndLayout

where $\tau \in \mathbb {H}$ with $\mathbb {H}:=\{\tau :\Im (\tau )>0\}$ denotes the Poincaré upper half-plane, and $\lambda _{\phi }(n)$ is the n-th normalized Fourier coefficient (Hecke eigenvalue), with the first Fourier coefficient normalized such that $\lambda _{\phi }(1)=1$ . From the theory of Hecke operators, it is well-known that $\lambda _{\phi }(n)\in \mathbb {R}$ for all $n\geqslant 1$ and also satisfies the multiplicative property

(1.1) $$ \begin{align} \lambda_{\phi}(m)\lambda_{\phi}(n) = \sum_{d|(m,n)} \lambda_{\phi}\bigg( \frac{mn}{d^{2}}\bigg), \end{align} $$ StartLayout 1st Row lamda Subscript phi Baseline left parenthesis m right parenthesis lamda Subscript phi Baseline left parenthesis n right parenthesis equals sigma summation Underscript d vertical bar left parenthesis m comma n right parenthesis Endscripts lamda Subscript phi Baseline left parenthesis StartFraction m n Over d squared EndFraction right parenthesis comma EndLayout

where $m,n\geqslant 1$ are any integers. The celebrated work of Deligne [Reference Deligne7] on Ramanujan–Petersson conjecture indicates that

$$ \begin{align*} |\lambda_{\phi}(n)| \leqslant d(n) \end{align*} $$

holds, where $d(n)$ is the classical divisor function.

The investigation of sign changes of modular forms is a classical and interesting topic, which attracts the focus of a handful of authors. By employing a classical theorem of Landau, coupled with certain analytic properties of the associated L-functions, Knopp et al. [Reference Knopp, Kohnen and Pribitkin30] established the result that $\{\lambda _{\phi }(n)\}_{n\geqslant 1}$ enjoys infinitely many sign changes, which can also be obtained from [Reference Murty44]. Afterward, there are a number of enlightening works devoted to the sign changes of Fourier coefficients of modular forms in various aspects (see, e.g., [Reference Hua16, Reference Iwaniec, Kohnen and Sengupta19, Reference Kohnen and Martin31, Reference Kumari and Murty33, Reference Lao and Luo35, Reference Matomäki42, Reference Vaishya52]). In particular, Banerjee and Pandey [Reference Banerjee and Pandey3] considered the sign changes of the sequence $\{\lambda _{\phi }(a^{2}+b^{2})\}_{a,b\geqslant 1}$ . In a more explicit fashion, they showed that $\{\lambda _{\phi }(a^{2}+b^{2})\}_{a,b\geqslant 1}$ has at least $\gg x^{\frac {1}{8}-2\varepsilon }$ sign changes in the interval $(x,2x]$ for sufficiently large x, for arbitrarily small $\varepsilon>0$ . Subsequently, this result has been improved and extended by Vaishya [Reference Vaishya53, Reference Vaishya54], using better subconvexity bounds for the lower-rank L-functions and more delicate analysis. In [Reference Mangerel41], Mangerel proved the remarkable fact that one can expect that $\{\lambda _{\phi }(a^{2}+b^{2})\}_{a,b\geqslant 1}$ has at most $\gg x(\log x)^{-\frac {1}{2}}$ sign changes in the interval $(x,2x]$ , which is the optimal result in this regard. In a more recent work, Lao and Qiao [Reference Lao and Qiao37] investigated the sign changes of the normalized coefficients of triple product L-functions in short intervals and also obtained the quantitative results over the indices represented by sums of two squares.

In a related framework, a number of authors also considered sign changes of Fourier coefficients of Hecke–Maass cusp forms. Let $M_{t}^{*}$ be the set of all normalized Hecke–Maass cusp form with Laplacian eigenvalue $\frac {1}{4}+t^{2}$ . Any $f\in M_{t}^{*}$ admits the Fourier expansion at $\infty $ :

$$ \begin{align*} f(\tau) = \sqrt{y}\sum_{n\neq 0} \lambda_{f}(n)K_{it}(2\pi |n|y)e(nx), \qquad \tau=x+iy\in\mathbb{H}, \end{align*} $$

where $K_{it}$ is the MacDonald–Bessel function, and $\lambda _{f}(n)$ is the n-th eigenvalue of the Hecke operators. Furthermore, the function $\lambda _{f}(n)$ is multiplicative and also satisfies the similar relation as that of (1.1) (cf. [Reference Hua13, Section 2]). While the Ramanujan conjecture for f remains widely open, the best record in this direction is due to Kim and Sarnak [Reference Kim28] who showed that

(1.2) $$ \begin{align} |\lambda_{f}(n)| \leqslant n^{\frac{7}{64}}d(n). \end{align} $$

In [Reference Knopp, Kohnen and Pribitkin30], the authors also mentioned that $\{\lambda _{f}(n)\}_{n\geqslant 1}$ has infinitely many sign changes, by adopting the overall similar strategy as in the holomorphic case. Take $f\in M_{t_{1}}^{*}$ and $g\in M_{t_{2}}^{*}$ be two distinct normalized Hecke–Maass cusp forms, and denote by $\lambda _{f}(n)$ and $\lambda _{g}(n)$ the n-th normalized Fourier coefficients of f and g, respectively. Throughout the article, we always assume f and g being two distinct normalized Hecke–Maass cusp forms. Kumari and Sengupta [Reference Kumari and Sengupta34] proved that the sequence $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n\geqslant 1}$ changes sign infinitely often, which further improved upon the works [Reference Gun, Kohnen and Rath9, Reference Kohnen and Sengupta32]. Furthermore, they also provided the explicit upper bound for first sign change of the $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n\geqslant 1}$ in terms of the spectral parameters and analytic conductors of the symmetric square L-functions associated with f and g. In a subsequent work, the author [Reference Hua13] established a quantitative result for the sign changes of the sequence $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n\geqslant 1}$ in short intervals, which significantly enhanced the conclusion in [Reference Kumari and Sengupta34]. Furthermore, in the same paper, the author also refined the upper bound of the first sign change of the same sequence comparable to [Reference Kumari and Sengupta34]. More recently, in the Hecke–Maass form setting, Kaur and Saha [Reference Kaur and Saha27] considered the number of sign changes of the sequence $\{\lambda _{f}(a^{2}+b^{2})\}_{a,b\in \mathbb {Z}}$ in short intervals. In a more explicit fashion, they showed that $\{\lambda _{f}(a^{2}+b^{2})\}_{a,b\in \mathbb {Z}}$ has at least $x^{1-\delta }$ sign changes in the interval $(x,2x]$ , provided that $\frac {3983}{4672}< \delta < 1$ . Moreover, in the remark of the same paper, they improved the result that the same sequence has at least $x^{1-\delta '}$ sign changes with $\frac {20651}{24384}< \delta '< 1$ , by utilizing the recent works on subconvexity bounds for $GL(2)$ and $GL(3) L$ -functions.

Inspired by the above enlightening and remarkable results, in the present article, we consider the quantitative version of the sign changes of the sequence $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n\geqslant 1}$ in the intervals $(x,2x]$ , which are supported at a certain integral binary quadratic forms. Before proceeding further, we introduce some basic terminology with respect to integral binary quadratic forms. Let $Q(x_{1},x_{2}):=ax_{1}^{2}+bx_{1}x_{2}+cx_{2}^{2}$ be an integral positive-definite binary quadratic form of discriminant $D=b^{2}-4ac$ , where $(x_{1},x_{2})\in \mathbb {Z}^{2}, a,b,c\in \mathbb {Z}$ . An integral binary quadratic form $Q(x_{1},x_{2})$ is called primitive if $(a,b,c)=1$ , and reduced if $|b|\leqslant a\leqslant c$ , and $b\geqslant 0$ if either $|b|=a$ or $a=c$ . Two forms $Q_{1}(x_{1},x_{2})$ and $Q_{2}(x_{1},x_{2})$ are called properly equivalent if there are integers $a,b,r,s$ with $as-br= 1$ such that $Q_{1}(x_{1},x_{2})=Q_{2}(ax_{1}+bx_{2}, rx_{1}+sx_{2})$ . For a given discriminant $D<0$ , let $\mathcal {S}_{D}$ denote the set of inequivalent primitive integral positive-definite binary quadratic reduced form of discriminant D, namely,

$$ \begin{align*} \mathcal{S}_{D} = \big\{ \mathcal{Q}: \mathcal{Q}~ \textrm{is a reduced form of discriminant}~ D\big\}. \end{align*} $$

The set $\mathcal {S}_{D}$ forms a group under certain composition law. Let $D<0$ be fixed. Then, the number of $h(D)$ of classes of primitive positive-definite forms of discriminant D is finite, and moreover $h(D)$ is equal to the number of reduced forms of discriminant D. For more details, we refer to [Reference Cox5, Chapter 2]. Now, we state our principal result as follows.

Theorem 1.1 Let $f\in M_{t_{1}}^{*}$ and $g\in M_{t_{2}}^{*}$ be distinct normalized Hecke–Maass cusp forms with Hecke eigenvalues $\{\lambda _{f}(n)\}$ and $\{\lambda _{g}(n)\}$ , respectively. Let $Q(\boldsymbol {x})\in \mathcal {S}_{D}$ be a certain quadratic form of fixed discriminant $D<0$ . For every sufficiently large $x\in \mathbb {R}$ , the sequence $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n=Q(\boldsymbol {x})}$ , as $\boldsymbol {x}=(x_{1},x_{2})$ varies in $\mathbb {Z}^{2}$ , has at least $\gg x^{1-\vartheta }$ sign changes in the interval $(x,2x]$ , where $\frac {31}{32}<\vartheta <1$ . In particular, the sequence $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n=Q(\boldsymbol {x})}$ enjoys infinitely many sign changes.

The structure of the current article is as follows. In Section 2, we gather some important and crucial properties of automorphic L-functions, along with the individual and averaged subconvexity/convexity bounds of L-functions available in the literature. Furthermore, we also give the decomposition of the associated L-series, which are related to the main result. In Section 3, we introduce the estimations of two prominent summatory functions, which are intimately linked with the proof of the main result. In Section 4, we complete the proof of Theorem $1.1$ , by using a general criterion of the recent work due to Lao and Qiao [Reference Lao and Qiao37], together with asymptotic formulas established in Section 3.

Notations. Throughout the article, the symbol $\varepsilon>0$ denotes an arbitrarily small positive constant for which the value may shift from one occurrence to another. We use Landau symbol $u=O(v)$ and Vinogradov symbol $u\ll v$ as synonyms, here, $u(x)\ll v(x)$ for all $x\in \mathfrak {X}$ (where the set $\mathfrak {X}$ has to be specified either explicitly or implicitly) denotes that $|u(x)|\leqslant Cv(x)$ for nonnegative function $v(x)$ and for some suitable constant $C>0$ . We always use p to denote a prime number, with or without subscript.

2 Preliminaries

In this section, we collect some basic properties of automorphic L-functions and also introduce a series of useful lemmas.

Let $f\in M_{t_{1}}^{*}$ be a normalized Hecke–Maass cusp form. For $\Re (s)>1$ , the Hecke L-function associated with f is defined as

$$ \begin{align*} L(f,s)= \prod_{p}\bigg( 1-\frac{\alpha_{f}(p)}{p^{s}}\bigg)^{-1}\bigg( 1-\frac{\beta_{f}(p)}{p^{s}}\bigg)^{-1} = \sum_{n=1}^{\infty} \frac{\lambda_{f}(n)}{n^{s}}, \end{align*} $$

where $\alpha _{f}(p),\beta _{f}(p)$ are the local parameters of f such that

(2.1) $$ \begin{align} \lambda_{f}(p) = \alpha_{f}(p) + \beta_{f}(p), \qquad \alpha_{f}(p) \beta_{f}(p) = 1. \end{align} $$

More generally, one can obtain

(2.2) $$ \begin{align} \lambda_{f}(p^{\nu}) = \lambda_{\mathrm{sym}^{\nu}f}(p) = \sum_{\ell=0}^{\nu} \alpha_{f}(p)^{\ell}\beta_{f}(p)^{\nu-\ell}, \qquad \nu\geqslant 1. \end{align} $$

The Kim–Sarnak bound is equivalent to the assertion that

(2.3) $$ \begin{align} |\alpha_{f}(p)| \leqslant p^{\frac{7}{64}}, \qquad |\beta_{f}(p)| \leqslant p^{\frac{7}{64}}. \end{align} $$

It is well-known that, since a primitive form (newform) f is associated with an automorphic cuspidal representation $\pi _{f}$ of $GL_{2}(\mathbb {A}_{\mathbb {Q}})$ , an automorphic L-function $L(\pi _{f},s)$ coincides with $L(f,s)$ (cf. [Reference Kim29]). In light of works of Gelbart–Jacquet [Reference Gelbart and Jacquet8] and Hoffstein–Lockhart [Reference Hoffstein and Lockhart11], for $\Re (s)>1$ , one can define the symmetric square L-function $L(\textrm {sym}^{2}f, s)$ associated with f as

$$ \begin{align*} L(\textrm{sym}^{2}f, s) = \prod_{p}\prod_{i=0}^{2} \bigg( 1-\frac{\alpha_{f}(p)^{2-i}\beta_{f}(p)^{i}}{p^{s}}\bigg)^{-1} = \sum_{n=1}^{\infty}\frac{\lambda_{\textrm{sym}^{2}f}(n)}{n^{s}}. \end{align*} $$

It is known that $L(\textrm {sym}^{2}f,s)$ is an entire function and also satisfies $L(\textrm {sym}^{2}f,1)\neq 0$ .

Let $\chi $ be a Dirichlet character modulo q. Then, in a similar pattern, for $j=1,2$ , one can define the twisted j-th symmetric power L-function via the Euler product of degree $j+1$

$$ \begin{align*} L(\textrm{sym}^{j}f\otimes\chi, s) &= \prod_{p}\prod_{i=0}^{j} \bigg( 1-\frac{\alpha_{f}(p)^{j-i}\beta_{f}(p)^{i}\chi(p)}{p^{s}}\bigg)^{-1} \nonumber \\ &= \sum_{n=1}^{\infty}\frac{\lambda_{\textrm{sym}^{j}f}(n)\chi(n)}{n^{s}}, \qquad \Re(s)>1. \end{align*} $$

StartLayout 1st Row 1st Column upper L left parenthesis sym Superscript j Baseline f circled times chi comma s right parenthesis 2nd Column equals product Underscript p Endscripts product Underscript i equals 0 Overscript j Endscripts left parenthesis 1 minus StartFraction alpha Subscript f Baseline left parenthesis p right parenthesis Superscript j minus i Baseline beta Subscript f Baseline left parenthesis p right parenthesis Superscript i Baseline chi left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction right parenthesis Superscript negative 1 Baseline 2nd Row 1st Column Blank 2nd Column equals sigma summation Underscript n equals 1 Overscript infinity Endscripts StartFraction lamda Subscript sym Sub Superscript j Subscript f Baseline left parenthesis n right parenthesis chi left parenthesis n right parenthesis Over n Superscript s Baseline EndFraction comma normal fraktur upper R left parenthesis s right parenthesis greater than 1 period EndLayout

Taking into account of the Rankin–Selberg convolution theory associated with two cuspidal automorphic representations developed by Jacquet et al. [Reference Jacquet, Piatetski and Shalika21], Jacquet and Shalika [Reference Jacquet and Shalika22, Reference Jacquet and Shalika23], Shahidi [Reference Shahidi48–Reference Shahidi51], and the reformulation of Rudnick and Sarnak [Reference Rudnick and Sarnak47], we can derive the analytic properties of $L(\pi \times \pi ',s)$ , where $\pi $ and $\pi '$ correspond to the known symmetric power lifting of two Hecke–Maass cusp forms. More explicitly, the Rankin–Selberg L-function $L(\textrm {sym}^{i}f\times \textrm {sym}^{j}g, s)$ attached to $\textrm {sym}^{i}f$ and $\textrm {sym}^{j}g$ with $1\leqslant i,j\leqslant 2$ is defined by

$$ \begin{align*} L(\textrm{sym}^{i}f\times \textrm{sym}^{j}g, s) = \prod_{p}\prod_{m=0}^{i}\prod_{m'=0}^{j}\big( 1-\alpha_{f}(p)^{i-2m}\alpha_{g}(p)^{j-2m'}p^{-s}\big)^{-1}, ~ \Re(s)>1. \end{align*} $$

For $\Re (s)>1$ , we may expand it into a Dirichlet series

$$ \begin{align*} L(\textrm{sym}^{i}f\times \textrm{sym}^{j}g, s) &= \prod_{p}\bigg( 1+ \sum_{r\geq 1}\frac{\lambda_{\textrm{sym}^{i}f\times\textrm{sym}^{j}g}(p^{r})}{p^{rs}}\bigg) \nonumber \\ &= \sum_{n=1}^{\infty}\frac{\lambda_{\textrm{sym}^{i}f\times\textrm{sym}^{j}g}(n)}{n^{s}}, \nonumber \end{align*} $$

StartLayout 1st Row 1st Column upper L left parenthesis sym Superscript i Baseline f times sym Superscript j Baseline g comma s right parenthesis 2nd Column equals product Underscript p Endscripts left parenthesis 1 plus sigma summation Underscript r greater than or equals 1 Endscripts StartFraction lamda Subscript sym Sub Superscript i Subscript f times sym Sub Superscript j Subscript g Baseline left parenthesis p Superscript r Baseline right parenthesis Over p Superscript r s Baseline EndFraction right parenthesis 2nd Row 1st Column Blank 2nd Column equals sigma summation Underscript n equals 1 Overscript infinity Endscripts StartFraction lamda Subscript sym Sub Superscript i Subscript f times sym Sub Superscript j Subscript g Baseline left parenthesis n right parenthesis Over n Superscript s Baseline EndFraction comma EndLayout

where the sequence $\{\lambda _{\textrm {sym}^{i}f\times \textrm {sym}^{j}g}(n)\}$ is real and multiplicative. In particular, one has $L(\textrm {sym}^{1}f\times \textrm {sym}^{1}g,s)=L(f\times g,s)$ and $L(\textrm {sym}^{1}f\times \textrm {sym}^{j}g,s)=L(f\times \textrm {sym}^{j}g,s)$ .

In a similar nature, for $\chi $ being a Dirichlet character with modulus q and $1\leqslant i,j\leqslant 2$ , one can also define the twisted Rankin–Selberg L-function

$$ \begin{align*} & L(\textrm{sym}^{i}f\times \textrm{sym}^{j}g\otimes\chi,s) \\ & \quad = \prod_{p}\prod_{m=0}^{i}\prod_{m'=0}^{j}\big( 1-\alpha_{f}(p)^{i-m}\beta_{f}(p)^{m}\alpha_{g}(p)^{j-m'}\beta_{g}(p)^{m'}\chi(p)p^{-s}\big)^{-1} \\ & \quad = \sum_{n=1}^{\infty}\frac{\lambda_{\textrm{sym}^{i}f\times\textrm{sym}^{j}g}(n)\chi(n)}{n^{s}}, \qquad \Re(s)>1. \end{align*} $$

StartLayout 1st Row 1st Column Blank 2nd Column upper L left parenthesis sym Superscript i Baseline f times sym Superscript j Baseline g circled times chi comma s right parenthesis 2nd Row 1st Column Blank 2nd Column equals product Underscript p Endscripts product Underscript m equals 0 Overscript i Endscripts product Underscript m prime equals 0 Overscript j Endscripts left parenthesis 1 minus alpha Subscript f Baseline left parenthesis p right parenthesis Superscript i minus m Baseline beta Subscript f Baseline left parenthesis p right parenthesis Superscript m Baseline alpha Subscript g Baseline left parenthesis p right parenthesis Superscript j minus m Super Superscript prime Superscript Baseline beta Subscript g Baseline left parenthesis p right parenthesis Superscript m Super Superscript prime Superscript Baseline chi left parenthesis p right parenthesis p Superscript negative s Baseline right parenthesis Superscript negative 1 Baseline 3rd Row 1st Column Blank 2nd Column equals sigma summation Underscript n equals 1 Overscript infinity Endscripts StartFraction lamda Subscript sym Sub Superscript i Subscript f times sym Sub Superscript j Subscript g Baseline left parenthesis n right parenthesis chi left parenthesis n right parenthesis Over n Superscript s Baseline EndFraction comma normal fraktur upper R left parenthesis s right parenthesis greater than 1 period EndLayout

It also inherits the properties similar to the Rankin–Selberg L-functions $L(\textrm {sym}^{i}f\times \textrm {sym}^{j}g, s)$ .

Let $r_{D}(n)$ denote the number of representations of $n\in \mathbb {Z}^{\geqslant 0}$ by the primitive integral positive-definite reduced binary quadratic forms of discriminant $D<0$ , namely,

$$ \begin{align*} \nonumber r_{D}(n):= \#\big\{ \boldsymbol{x}\in\mathbb{Z}^{2}~:~ n=\mathcal{Q}(\boldsymbol{x}), ~\mathcal{Q}\in\mathcal{S}_{D}\big\}. \end{align*} $$

In the literature, the function $r_{D}(n)$ is commonly referred to as character sum. The generating function $\theta _{D}(\tau )$ for the character sum $r_{D}(n)$ is defined as

$$ \begin{align*} \nonumber \theta_{D}(\tau) := \sum_{n=0}^{\infty} r_{D}(n)q^{n} =\sum_{\mathcal{Q}\in\mathcal{S}_{D}, \boldsymbol{x}\in\mathbb{Z}^{2}} q^{\mathcal{Q}(\boldsymbol{x})}, \qquad q := e^{2\pi i\tau}. \end{align*} $$

From [Reference Iwaniec18, Theorem 10.9], we learn that $\theta _{D}(\tau ) \in M_{1}(\Gamma _{0}(|D|),\chi _{D})$ , where $\chi _{D}$ is the Dirichlet character modulo $|D|$ which can be represented by the Kronecker symbol $\chi _{D}(d):=(\frac {D}{d})$ . Here, we denote by $M_{1}(\Gamma _{0}(|D|),\chi _{D})$ the $\mathbb {C}$ -vector space of modular forms of weight $1$ , level $|D|$ , and nebentypus $\chi _{D}$ . By appealing to [Reference Iwaniec18, Section 11.2], we also know that

$$ \begin{align*} \nonumber r_{D}(n) = w_{D}\sum_{d|n}\chi_{D}(d), \qquad w_{D}= \left\{ \begin{array}{@{}ll} 6, & \quad \textrm{D} = -3,\\ 4, & \quad \textrm{D} = -4, \\ 2, & \quad \textrm{D} < -4, \end{array} \right. \end{align*} $$

r Subscript upper D Baseline left parenthesis n right parenthesis equals w Subscript upper D Baseline sigma summation Underscript d vertical bar n Endscripts chi Subscript upper D Baseline left parenthesis d right parenthesis comma w Subscript upper D Baseline equals StartLayout Enlarged left brace 1st Row 1st Column 6 comma 2nd Column upper D equals negative 3 comma 2nd Row 1st Column 4 comma 2nd Column upper D equals negative 4 comma 3rd Row 1st Column 2 comma 2nd Column upper D less than negative 4 comma EndLayout

where $w_{D}$ is the size of the group of units. The formula for $r_{D}(n)$ depends only on the discriminant D, and not rely on the choice of the forms appearing in the class group $\mathcal {S}_{D}$ corresponding to discriminant D. Moreover, the well-known Weil’s bound reveals $r_{D}(n)\ll n^{\varepsilon }$ (cf. [Reference Hu, Jiang and Lü12, (1.1)] and references therein).

For brevity, we set $r_{D}(n)= w_{D} r_{D}^{*}(n)$ , where

$$ \begin{align*} \nonumber r_{D}^{*}(n) := \sum_{d|n}\chi_{D}(d). \end{align*} $$

Specifically, one can obtain

(2.4) $$ \begin{align} r_{D}^{*}(p) = 1+ \chi_{D}(p), \qquad r_{D}^{*}(p^{2}) = 1+\chi_{D}(p) + \chi_{D}(p^{2}). \end{align} $$

Now, we record some subconvexity and convexity bounds for the associated L-functions in the critical strip, which play a significant role for our purpose.

Lemma 2.1 For any $\varepsilon>0$ , we have

$$ \begin{align*} \nonumber \zeta(\sigma+it) \ll \big( |t| + 1\big)^{\max\{\frac{13}{42}(1-\sigma),0\}+\varepsilon}, \end{align*} $$

uniformly for $\frac {1}{2}\leqslant \sigma \leqslant 1+\varepsilon $ and $|t|\geqslant 1$ .

Proof This result follows from [Reference Bourgain4, Theorem 5] and Phragmén–Lindelöf convexity principle for a strip [Reference Iwaniec and Kowalski20, Theorem 5.53].

Lemma 2.2 Let $f\in M_{t_{1}}^{*}$ be a normalized Hecke–Maass cusp form. For any $\varepsilon>0$ , we have

$$ \begin{align*} \nonumber L(\mathrm{sym}^{2} f,\sigma+it) \ll \big( |t| + 1\big)^{\max\{\frac{8}{7}(1-\sigma),0\}+\varepsilon}, \end{align*} $$

uniformly for $\frac {1}{2}\leqslant \sigma \leqslant 1+\varepsilon $ and $|t|\geqslant 1$ .

Proof This result is a straightforward deduction from the recent work of Dasgupta et al. [Reference Dasgupta, Leung and Young6, Corollary 1.4].

Lemma 2.3 Let $f\in M_{t_{1}}^{*}$ be a normalized Hecke–Maass cusp form, and let $\chi _{D}$ be a Dirichlet character modulo $|D|$ . For any $\varepsilon>0$ , we have

$$ \begin{align*} \nonumber L(\chi_{D}, \sigma+it) \ll \big( |t| + 1\big)^{\max\{\frac{13}{42}(1-\sigma),0\}+\varepsilon}, \nonumber \end{align*} $$

and

$$ \begin{align*} \nonumber L(\mathrm{sym}^{2}f\otimes\chi_{D},\sigma+it) \ll \big( |t| + 1\big)^{\max\{\frac{8}{7}(1-\sigma),0\}+\varepsilon}, \end{align*} $$

uniformly for $\frac {1}{2}\leqslant \sigma \leqslant 1+\varepsilon $ and $|t|\geqslant 1$ .

Proof Following the argument of [Reference Kaur and Saha27, Lemma 2.2] with slight modifications, along with appealing to Lemmas 2.1 and 2.2, one sees that twisting the L-functions by a Dirichlet character $\chi $ does not alter the subconvexity bounds with respect to the t aspect.

Let $\boldsymbol {d} := (d_{1},\ldots ,d_{J})$ , $\boldsymbol {m} = (m_{1},\ldots ,m_{J}),$ and $\boldsymbol {n} = (n_{1},\ldots ,n_{J})$ with $d_{j},m_{j},n_{j}\in (\mathbb {N}\cup \{0\})$ , and set

$$ \begin{align*} \nonumber A(\boldsymbol{d}, \boldsymbol{m},\boldsymbol{n}) = \frac{1}{2}\sum_{j=1}^{J}d_{j}(m_{j}+1)(n_{j}+1). \end{align*} $$ upper A left parenthesis bold italic d comma bold italic m comma bold italic n right parenthesis equals one half sigma summation Underscript j equals 1 Overscript upper J Endscripts d Subscript j Baseline left parenthesis m Subscript j Baseline plus 1 right parenthesis left parenthesis n Subscript j Baseline plus 1 right parenthesis period

Let $\chi $ be a primitive character modulo q, and define

(2.5) $$ \begin{align} \mathfrak{L}_{\boldsymbol{m},\boldsymbol{n}}^{\boldsymbol{d}}(f,g,\chi,s) := \prod_{j=1}^{J}L(\textrm{sym}^{m_{j}}f\times \textrm{sym}^{n_{j}}g\otimes\chi, s)^{d_{j}}. \end{align} $$

The function $\mathfrak {L}_{\boldsymbol {m},\boldsymbol {n}}^{\boldsymbol {d}}(f,g,\chi ,s)$ is an example of a general L-functions in the sense of Perelli [Reference Perelli45]. We also have the following convexity bound.

Lemma 2.4 For any $\varepsilon>0$ , we have

(2.6) $$ \begin{align} \int_{T}^{2T} \big| \mathfrak{L}_{\boldsymbol{m},\boldsymbol{n}}^{\boldsymbol{d}}(f,g,\chi,\sigma+it) \big|^{2} dt \ll \big( q(T+1)\big)^{\max\{2A(\boldsymbol{d},\boldsymbol{m},\boldsymbol{n})(1-\sigma),1\}+\varepsilon} \end{align} $$

uniformly for $\frac {1}{2}\leqslant \sigma \leqslant 2$ and $T\geqslant 1$ .

Proof The results follow readily from Perelli [Reference Perelli45, Theorem 4], and the interested reader can also refer to Jiang and Lü [Reference Jiang and Lü24, Lemma 2.4] (see also [Reference Hua17, Lemma 2.7, Remark 2.8]).

To make our proof of the main result transparent and accessible, we define two summatory functions as follows:

(2.7) $$ \begin{align} S(x; f,g, D) := \sum_{\substack{\mathcal{Q}(\boldsymbol{x}) \leqslant x \\ \mathcal{Q}\in \mathcal{S}_{D},\boldsymbol{x}\in\mathbb{Z}^{2}}} \lambda_{f}(\mathcal{Q}(\boldsymbol{x}))\lambda_{g}(\mathcal{Q}(\boldsymbol{x})) = \sum_{n\leqslant x} \lambda_{f}(n)\lambda_{g}(n)r_{D}(n), \end{align} $$

and

(2.8) $$ \begin{align} T(x;f,g, D) := \sum_{\substack{\mathcal{Q}(\boldsymbol{x}) \leqslant x \\ \mathcal{Q}\in \mathcal{S}_{D},\boldsymbol{x}\in\mathbb{Z}^{2}}} \lambda_{f}^{2}(\mathcal{Q}(\boldsymbol{x}))\lambda_{g}^{2}(\mathcal{Q}(\boldsymbol{x})) = \sum_{n\leqslant x} \lambda_{f}^{2}(n)\lambda_{g}^{2}(n)r_{D}(n). \end{align} $$

To detect the sign changes of the sequence $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n=Q(\boldsymbol {x})}$ in the interval $(x,2x]$ , for $\Re (s)> 1$ , it is essential to consider the following two L-series:

(2.9) $$ \begin{align} \mathfrak{F}(s) := \sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\lambda_{g}(n)r_{D}(n)}{n^{s}} = w_{D}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)\lambda_{g}(n)r_{D}^{*}(n)}{n^{s}}, \end{align} $$

and

(2.10) $$ \begin{align} \mathfrak{G}(s) := \sum_{n=1}^{\infty}\frac{\lambda_{f}^{2}(n)\lambda_{g}^{2}(n)r_{D}(n)}{n^{s}} = w_{D}\sum_{n=1}^{\infty}\frac{\lambda_{f}^{2}(n)\lambda_{g}^{2}(n)r_{D}^{*}(n)}{n^{s}}. \end{align} $$

In the next two lemmas, we provide the decompositions of the associated L-series $\mathfrak {F}(s)$ and $\mathfrak {G}(s)$ .

Lemma 2.5 Let $f\in M_{t_{1}}^{*}$ and $g\in M_{t_{2}}^{*}$ be two distinct normalized Hecke–Maass cusp forms, and let $\mathfrak {F}(s)$ be defined as (2.9). Then,

(2.11) $$ \begin{align} \mathfrak{F}(s) = L(f\times g, s)L(f\times g\otimes\chi_{D}, s)U(s), \end{align} $$

where $U(s)$ is a Dirichlet series which converges absolutely and uniformly for $\Re (s)> \frac {1}{2}$ , and $U(s) \neq 0$ for $\Re (s)=1$ .

Proof Using the multiplicative property of $\lambda _{f}(n)\lambda _{g}(n)$ and $r_{D}^{*}(n)$ , for $\Re (s)>1$ , one has the Euler product expansion

(2.12) $$ \begin{align} \mathfrak{F}(s) = w_{D}\mathfrak{F}_{1}(s), \quad \mathrm{where}\quad \mathfrak{F}_{1}(s)= \prod_{p}\bigg( 1+ \sum_{j\geqslant 1} \frac{\lambda_{f}(p^{j})\lambda_{g}(p^{j})r_{D}^{*}(p^{j})}{p^{js}}\bigg). \end{align} $$

In view of (2.12), one can also rewrite $\mathfrak {F}_{1}(s)$ as

$$ \begin{align*} \mathfrak{F}_{1}(s) &= \exp\bigg\{ \sum_{p}\log\bigg( 1+\frac{\lambda_{f}(p)\lambda_{g}(p)r_{D}^{*}(p)}{p^{s}} + \frac{\lambda_{f}(p^{2})\lambda_{g}(p^{2})r_{D}^{*}(p^{2})}{p^{2s}} \nonumber \\ &\quad + \cdots+ \frac{\lambda_{f}(p^{\nu})\lambda_{g}(p^{\nu})r_{D}^{*}(p^{\nu})}{p^{\nu s}}+ \cdots\bigg)\bigg\}. \nonumber \end{align*} $$

StartLayout 1st Row 1st Column German upper F 1 left parenthesis s right parenthesis 2nd Column equals exp left brace sigma summation Underscript p Endscripts log left parenthesis 1 plus StartFraction lamda Subscript f Baseline left parenthesis p right parenthesis lamda Subscript g Baseline left parenthesis p right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction plus StartFraction lamda Subscript f Baseline left parenthesis p squared right parenthesis lamda Subscript g Baseline left parenthesis p squared right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p squared right parenthesis Over p Superscript 2 s Baseline EndFraction 2nd Row 1st Column Blank 2nd Column plus midline horizontal ellipsis plus StartFraction lamda Subscript f Baseline left parenthesis p Superscript nu Baseline right parenthesis lamda Subscript g Baseline left parenthesis p Superscript nu Baseline right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p Superscript nu Baseline right parenthesis Over p Superscript nu s Baseline EndFraction plus midline horizontal ellipsis right parenthesis right brace period EndLayout

Using the Taylor expansion for $\log (1+z)$ , we may deduce that

(2.13) $$ \begin{align} \log \mathfrak{F}_{1}(s) &= \!\sum_{p}\bigg(\! \frac{\lambda_{f}(p)\lambda_{g}(p)r_{D}^{*}(p)}{p^{s}} + \frac{\lambda_{f}(p^{2})\lambda_{g}(p^{2})r_{D}^{*}(p^{2})- \frac{1}{2}\lambda_{f}^{2}(p)\lambda_{g}^{2}(p)(r_{D}^{*}(p))^{2}}{p^{2s}} \nonumber \\ &\quad + \sum_{\ell\geqslant 3} \frac{h(p^{\ell})}{p^{\ell s}}\bigg), \end{align} $$

StartLayout 1st Row 1st Column log German upper F 1 left parenthesis s right parenthesis 2nd Column equals sigma summation Underscript p Endscripts left parenthesis StartFraction lamda Subscript f Baseline left parenthesis p right parenthesis lamda Subscript g Baseline left parenthesis p right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction plus StartFraction lamda Subscript f Baseline left parenthesis p squared right parenthesis lamda Subscript g Baseline left parenthesis p squared right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p squared right parenthesis minus one half lamda Subscript f Superscript 2 Baseline left parenthesis p right parenthesis lamda Subscript g Superscript 2 Baseline left parenthesis p right parenthesis left parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis right parenthesis squared Over p Superscript 2 s Baseline EndFraction 2nd Row 1st Column Blank 2nd Column plus sigma summation Underscript script l greater than or slanted equals 3 Endscripts StartFraction h left parenthesis p Superscript script l Baseline right parenthesis Over p Superscript script l s Baseline EndFraction right parenthesis comma EndLayout

where $h(p^{\ell }),(\ell \geqslant 3)$ can be determined explicitly, but is rather tedious and requires laborious work, which is not necessarily needed in our context. From (2.4), it is easy to see that

(2.14) $$ \begin{align} \lambda_{f}(p)\lambda_{g}(p)r_{D}^{*}(p) = \lambda_{f}(p)\lambda_{g}(p)(1+\chi_{D}(p)) = \lambda_{f\times g}(p)(1+\chi_{D}(p)). \end{align} $$

Moreover, using the relations (2.1), (2.2) and (2.4), we learn that

(2.15) $$ \begin{align} &\lambda_{f}(p^{2})\lambda_{g}(p^{2})r_{D}^{*}(p^{2}) \nonumber \\ &\quad = (\alpha_{f}^{2}(p) + 1+ \beta_{f}^{2}(p))(\alpha_{g}^{2}(p) + 1+ \beta_{g}^{2}(p))(1+ \chi_{D}(p) + \chi_{D}(p^{2})), \end{align} $$

and

(2.16) $$ \begin{align} & \lambda_{f}^{2}(p)\lambda_{g}^{2}(p)(r_{D}^{*}(p))^{2}\nonumber \\ &\quad = (1+\lambda_{f}(p^{2}))(1+\lambda_{g}(p^{2}))(1+\chi_{D}(p))^{2} \nonumber \\ &\quad = \big(2+\alpha_{f}^{2}(p) + \beta_{f}^{2}(p) \big)\big(2+\alpha_{g}^{2}(p) + \beta_{g}^{2}(p) \big)(1+2\chi_{D}(p) + \chi_{D}(p^{2})), \end{align} $$

StartLayout 1st Row 1st Column Blank 2nd Column lamda Subscript f Superscript 2 Baseline left parenthesis p right parenthesis lamda Subscript g Superscript 2 Baseline left parenthesis p right parenthesis left parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis right parenthesis squared 2nd Row 1st Column Blank 2nd Column equals left parenthesis 1 plus lamda Subscript f Baseline left parenthesis p squared right parenthesis right parenthesis left parenthesis 1 plus lamda Subscript g Baseline left parenthesis p squared right parenthesis right parenthesis left parenthesis 1 plus chi Subscript upper D Baseline left parenthesis p right parenthesis right parenthesis squared 3rd Row 1st Column Blank 2nd Column equals left parenthesis 2 plus alpha Subscript f Superscript 2 Baseline left parenthesis p right parenthesis plus beta Subscript f Superscript 2 Baseline left parenthesis p right parenthesis right parenthesis left parenthesis 2 plus alpha Subscript g Superscript 2 Baseline left parenthesis p right parenthesis plus beta Subscript g Superscript 2 Baseline left parenthesis p right parenthesis right parenthesis left parenthesis 1 plus 2 chi Subscript upper D Baseline left parenthesis p right parenthesis plus chi Subscript upper D Baseline left parenthesis p squared right parenthesis right parenthesis comma EndLayout

respectively. Henceforth, by substituting (2.14)–(2.16) back into (2.13), one can readily obtain

(2.17) $$ \begin{align} \log \mathfrak{F}_{1}(s) = \sum_{p} \frac{h(p)}{p^{s}} + \sum_{p}\frac{h(p^{2})}{p^{2s}} + \sum_{p}\sum_{\ell\geqslant 3}\frac{h(p^{\ell})}{p^{\ell s}}, \end{align} $$

where

(2.18) $$ \begin{align} h(p) := \lambda_{f\times g}(p)(1+\chi_{D}(p)), \end{align} $$

and

(2.19) $$ \begin{align} h(p^{2}) &:= \lambda_{f}(p^{2})\lambda_{g}(p^{2})r_{D}^{*}(p^{2})- \frac{1}{2}\lambda_{f}^{2}(p)\lambda_{g}^{2}(p)(r_{D}^{*}(p))^{2} \nonumber \\ &= \frac{1}{2}\big( \lambda_{f}(p^{2})\lambda_{g}(p^{2}) (1+\chi_{D}(p^{2})) - (1+\lambda_{f}(p^{2}) + \lambda_{g}(p^{2}))r_{D}^{*}(p)^{2}\big), \end{align} $$

StartLayout 1st Row 1st Column h left parenthesis p squared right parenthesis 2nd Column colon equals lamda Subscript f Baseline left parenthesis p squared right parenthesis lamda Subscript g Baseline left parenthesis p squared right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p squared right parenthesis minus one half lamda Subscript f Superscript 2 Baseline left parenthesis p right parenthesis lamda Subscript g Superscript 2 Baseline left parenthesis p right parenthesis left parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis right parenthesis squared 2nd Row 1st Column Blank 2nd Column equals one half left parenthesis lamda Subscript f Baseline left parenthesis p squared right parenthesis lamda Subscript g Baseline left parenthesis p squared right parenthesis left parenthesis 1 plus chi Subscript upper D Baseline left parenthesis p squared right parenthesis right parenthesis minus left parenthesis 1 plus lamda Subscript f Baseline left parenthesis p squared right parenthesis plus lamda Subscript g Baseline left parenthesis p squared right parenthesis right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis squared right parenthesis comma EndLayout

respectively. By analogy, the coefficients $h(p^{\ell })$ can also be represented as the linear combinations of Hecke eigenvalues on higher power prime arguments of the similar flavor as above, but we do not necessarily need the precise expressions in our context. Instead, we merely need the upper bound $h(p^{\ell })\ll p^{\frac {7}{32}\ell }$ , due to the bound of Kim–Sarnak (2.3). From (2.17), we infer that the analytic properties of $\mathfrak {F}_{1}(s)$ are determined by the coefficient of $p^{-s}$ such that

(2.20) $$ \begin{align} \mathcal{F}(s) := L(f\times g,s)L(f\times g\otimes\chi_{D},s). \end{align} $$

We verify that

(2.21) $$ \begin{align} \log L(f\times g, s) &= -\sum_{p} \bigg\{ \log\bigg( 1-\frac{\alpha_{f}(p)\alpha_{g}(p)}{p^{s}}\bigg) + \log\bigg( 1-\frac{\beta_{f}(p)\alpha_{g}(p)}{p^{s}}\bigg)\nonumber \\ &\quad + \log\bigg( 1-\frac{\alpha_{f}(p)\beta_{g}(p)}{p^{s}}\bigg)+ \log\bigg( 1-\frac{\beta_{f}(p)\beta_{g}(p)}{p^{s}}\bigg)\bigg\}\nonumber \\ &= \sum_{p} \bigg( \frac{\lambda_{f\times g}(p)}{p^{s}} + \frac{w(p^{2})}{p^{2s}} + \cdots\bigg), \end{align} $$

StartLayout 1st Row 1st Column log upper L left parenthesis f times g comma s right parenthesis 2nd Column equals minus sigma summation Underscript p Endscripts left brace log left parenthesis 1 minus StartFraction alpha Subscript f Baseline left parenthesis p right parenthesis alpha Subscript g Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction right parenthesis plus log left parenthesis 1 minus StartFraction beta Subscript f Baseline left parenthesis p right parenthesis alpha Subscript g Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction right parenthesis 2nd Row 1st Column Blank 2nd Column plus log left parenthesis 1 minus StartFraction alpha Subscript f Baseline left parenthesis p right parenthesis beta Subscript g Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction right parenthesis plus log left parenthesis 1 minus StartFraction beta Subscript f Baseline left parenthesis p right parenthesis beta Subscript g Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction right parenthesis right brace 3rd Row 1st Column Blank 2nd Column equals sigma summation Underscript p Endscripts left parenthesis StartFraction lamda Subscript f times g Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction plus StartFraction w left parenthesis p squared right parenthesis Over p Superscript 2 s Baseline EndFraction plus midline horizontal ellipsis right parenthesis comma EndLayout

where

$$ \begin{align*} \nonumber w(p^{2}) = \frac{1}{2}\big( \alpha_{f}^{2}(p) + \beta_{f}^{2}(p)\big)\big( \alpha_{g}^{2}(p) + \beta_{g}^{2}(p)\big)= \frac{1}{2}\big( \lambda_{f}(p^{2}) - 1\big) \big( \lambda_{g}(p^{2}) - 1\big). \nonumber \end{align*} $$

Analogously, one can also obtain

(2.22) $$ \begin{align} \log L(f\times g\otimes\chi_{D}, s) = \sum_{p} \bigg( \frac{\lambda_{f\times g}(p)\chi_{D}(p)}{p^{s}} + \frac{w^{*}(p^{2})}{p^{2s}} + \cdots\bigg), \end{align} $$

where

$$ \begin{align*} \nonumber w^{*}(p^{2}) = \frac{1}{2}\big( \lambda_{f}(p^{2}) - 1\big) \big( \lambda_{g}(p^{2}) - 1\big)\chi_{D}(p^{2}). \end{align*} $$

Putting together (2.20)–(2.22), it follows that

(2.23) $$ \begin{align} \log \mathcal{F}(s)= \sum_{p} \bigg( \frac{h(p)}{p^{s}} + \frac{V(p^{2})}{p^{2s}} + \sum_{\ell\geqslant 3} \frac{V(p^{\ell})}{p^{\ell s}}\bigg), \end{align} $$

where $h(p)$ is defined as (2.18), and

(2.24) $$ \begin{align} V(p^{2})= \frac{1}{2}\big( \lambda_{f}(p^{2}) - 1\big) \big( \lambda_{g}(p^{2}) - 1\big)\big( 1+ \chi_{D}(p^{2})\big). \end{align} $$

Combining (2.12), (2.17) and (2.24), we reinterpret $\mathfrak {F}(s)$ as

$$ \begin{align*} \nonumber \mathfrak{F}(s) &= w_{D}\mathcal{F}(s)\cdot \exp\bigg\{ \sum_{p} \bigg( \frac{h(p^{2}) - V(p^{2})}{p^{2s}} + \sum_{\ell\geqslant 3} \frac{h(p^{\ell}) - V(p^{\ell})}{p^{\ell s}}\bigg)\bigg\}\nonumber \\ &= \mathcal{F}(s)\cdot w_{D}\prod_{p}\bigg( 1+\frac{h(p^{2})-V(p^{2})}{p^{2s}} + \cdots\bigg) \nonumber \\ &:= \mathcal{F}(s)U(s), \nonumber \end{align*} $$

StartLayout 1st Row 1st Column German upper F left parenthesis s right parenthesis 2nd Column equals w Subscript upper D Baseline script upper F left parenthesis s right parenthesis dot exp left brace sigma summation Underscript p Endscripts left parenthesis StartFraction h left parenthesis p squared right parenthesis minus upper V left parenthesis p squared right parenthesis Over p Superscript 2 s Baseline EndFraction plus sigma summation Underscript script l greater than or slanted equals 3 Endscripts StartFraction h left parenthesis p Superscript script l Baseline right parenthesis minus upper V left parenthesis p Superscript script l Baseline right parenthesis Over p Superscript script l s Baseline EndFraction right parenthesis right brace 2nd Row 1st Column Blank 2nd Column equals script upper F left parenthesis s right parenthesis dot w Subscript upper D Baseline product Underscript p Endscripts left parenthesis 1 plus StartFraction h left parenthesis p squared right parenthesis minus upper V left parenthesis p squared right parenthesis Over p Superscript 2 s Baseline EndFraction plus midline horizontal ellipsis right parenthesis 3rd Row 1st Column Blank 2nd Column colon equals script upper F left parenthesis s right parenthesis upper U left parenthesis s right parenthesis comma EndLayout

where

$$ \begin{align*} \nonumber h(p^{2}) - V(p^{2}) &= -\big(\lambda_{f}(p^{2}) + \lambda_{g}(p^{2})\big)\chi_{D}(p) - r_{D}^{*}(p^{2})\nonumber \\ &= -\big( 2+\alpha_{f}^{2}(p)+ \beta_{f}^{2}(p)+ \alpha_{g}^{2}(p)+ \beta_{g}^{2}(p)\big)\chi_{D}(p)\\ &\quad - (1+\chi_{D}(p) + \chi_{D}(p^{2})). \nonumber \end{align*} $$

StartLayout 1st Row 1st Column h left parenthesis p squared right parenthesis minus upper V left parenthesis p squared right parenthesis 2nd Column equals minus left parenthesis lamda Subscript f Baseline left parenthesis p squared right parenthesis plus lamda Subscript g Baseline left parenthesis p squared right parenthesis right parenthesis chi Subscript upper D Baseline left parenthesis p right parenthesis minus r Subscript upper D Superscript asterisk Baseline left parenthesis p squared right parenthesis 2nd Row 1st Column Blank 2nd Column equals minus left parenthesis 2 plus alpha Subscript f Superscript 2 Baseline left parenthesis p right parenthesis plus beta Subscript f Superscript 2 Baseline left parenthesis p right parenthesis plus alpha Subscript g Superscript 2 Baseline left parenthesis p right parenthesis plus beta Subscript g Superscript 2 Baseline left parenthesis p right parenthesis right parenthesis chi Subscript upper D Baseline left parenthesis p right parenthesis 3rd Row 1st Column Blank 2nd Column minus left parenthesis 1 plus chi Subscript upper D Baseline left parenthesis p right parenthesis plus chi Subscript upper D Baseline left parenthesis p squared right parenthesis right parenthesis period EndLayout

From [Reference Lao, McKee and Ye36, Lemma 2.2], for $\sigma>1$ , we know that

(2.25) $$ \begin{align} \prod_{p} \bigg( 1+\frac{|\alpha_{\pi}(p)|^{\nu} + |\beta_{\pi}(p)|^{\nu}}{p^{\sigma}}\bigg) \ll 1 \end{align} $$

for $0\leqslant \nu \leqslant 8$ . Here, $\pi =\otimes _{p}\pi _{p}$ denotes the unitary automorphic irreducible cuspidal representation of $GL_{2}(\mathbb {A}_{\mathbb {Q}})$ with Fourier coefficients $\{\lambda _{\pi }(n)\}$ , and $\alpha _{\pi }(p), \beta _{\pi }(p)\in \mathbb {C}$ are the Satake parameters associated with $\lambda _{\pi }(p)$ . Henceforth, we obtain

$$ \begin{align*} \nonumber U(s) \ll \prod_{p} \bigg( 1+\frac{|\alpha_{f}(p)|^{2} + |\beta_{f}(p)|^{2}}{p^{2\sigma}}\bigg) \ll 1, \end{align*} $$

for any $\sigma>\frac {1}{2}$ , which implies that $U(s)$ converges absolutely for $\Re (s)>\frac {1}{2}$ .

For the non-vanishing of $U(s) \neq 0$ on $\Re (s)=1$ , one can apply the approach described in [Reference Michel, Sarnak and Shahidi43, Section 1.2]. To be more precise, one can consider the auxiliary product

$$ \begin{align*} \nonumber D(s):= \zeta(s)\mathfrak{F}(s+it_{0})^{2}\mathfrak{F}(s+2it_{0}). \end{align*} $$

The above Dirichlet series has nonnegative coefficients, and convergent for $s>1$ . Following an extension of Hadamard and de la Vallée-Poussin method, if $\mathfrak {F}(1+it_{0})=0$ , then $D(s)$ has no pole at $s=1$ , contradicting Landau’s lemma. Henceforth, we can in turn deduce that $U(s) \neq 0$ on $\Re (s)=1$ .

Lemma 2.6 Let $f\in M_{t_{1}}^{*}$ and $g\in M_{t_{2}}^{*}$ be two distinct normalized Hecke–Maass cusp forms, and let $\mathfrak {G }(s)$ be defined as (2.10). Then,

(2.26) $$ \begin{align} \mathfrak{G}(s) &= \zeta(s)L(\chi_{D}, s)L(\mathrm{sym}^{2}f, s)L(\mathrm{sym}^{2}g, s)L(\mathrm{sym}^{2}f\times \mathrm{sym}^{2}g, s)\nonumber \\&\quad \cdot L(\mathrm{sym}^{2}f\otimes\chi_{D}, s)L(\mathrm{sym}^{2}g\otimes\chi_{D}, s)L(\mathrm{sym}^{2}f\times \mathrm{sym}^{2}g\otimes\chi_{D}, s)V(s), \end{align} $$

where $V(s)$ is a Dirichlet series which converges absolutely and uniformly for $\Re (s)> \frac {1}{2}$ , and $V(s) \neq 0$ for $\Re (s)=1$ .

Proof The proof follows a similar argument as that of Lemma $2.5$ , and we only sketch the proof briefly. By the property of multiplicativity of $\lambda _{f}^{2}(n)\lambda _{g}^{2}(n)r_{D}^{*}(n)$ , for $\Re (s)>1$ , one can attain

(2.27) $$ \begin{align} \mathfrak{G}(s) = w_{D}\mathfrak{G}_{1}(s), \quad \mathrm{where} \quad \mathfrak{G}_{1}(s):= \prod_{p}\bigg( 1+ \sum_{j\geqslant 1} \frac{\lambda_{f}^{2}(p^{j})\lambda_{g}^{2}(p^{j})r_{D}^{*}(p^{j})}{p^{js}}\bigg). \end{align} $$

By virtue of (2.27), one can also rephrase $\mathfrak {G}_{1}(s)$ as

$$ \begin{align*} \nonumber \mathfrak{G}_{1}(s) &= \exp\bigg\{ \sum_{p}\log\bigg( 1+\frac{\lambda_{f}^{2}(p)\lambda_{g}^{2}(p)r_{D}^{*}(p)}{p^{s}} + \frac{\lambda_{f}^{2}(p^{2})\lambda_{g}^{2}(p^{2})r_{D}^{*}(p^{2})}{p^{2s}} \nonumber \\ &\quad + \cdots+ \frac{\lambda_{f}^{2}(p^{\nu})\lambda_{g}^{2}(p^{\nu})r_{D}^{*}(p^{\nu})}{p^{\nu s}}+ \cdots\bigg)\bigg\}. \nonumber \end{align*} $$

StartLayout 1st Row 1st Column German upper G 1 left parenthesis s right parenthesis 2nd Column equals exp left brace sigma summation Underscript p Endscripts log left parenthesis 1 plus StartFraction lamda Subscript f Superscript 2 Baseline left parenthesis p right parenthesis lamda Subscript g Superscript 2 Baseline left parenthesis p right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction plus StartFraction lamda Subscript f Superscript 2 Baseline left parenthesis p squared right parenthesis lamda Subscript g Superscript 2 Baseline left parenthesis p squared right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p squared right parenthesis Over p Superscript 2 s Baseline EndFraction 2nd Row 1st Column Blank 2nd Column plus midline horizontal ellipsis plus StartFraction lamda Subscript f Superscript 2 Baseline left parenthesis p Superscript nu Baseline right parenthesis lamda Subscript g Superscript 2 Baseline left parenthesis p Superscript nu Baseline right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p Superscript nu Baseline right parenthesis Over p Superscript nu s Baseline EndFraction plus midline horizontal ellipsis right parenthesis right brace period EndLayout

From [Reference Lau and Lü38, Lemma 7.1], we learn that

(2.28) $$ \begin{align} \lambda_{f}^{4}(p)= 2+ 3\lambda_{\textrm{sym}^{2}f}(p) + \lambda_{\textrm{sym}^{4}f}(p). \end{align} $$

Using the Taylor expansion for $\log (1+z)$ , we may deduce that

(2.29) $$ \begin{align} \log \mathfrak{G}_{1}(s) = \sum_{p}\bigg( \frac{\lambda_{f}^{2}(p)\lambda_{g}^{2}(p)r_{D}^{*}(p)}{p^{s}} + \frac{A(p^{2})}{p^{2s}} + \sum_{\ell\geqslant 3} \frac{A(p^{\ell})}{p^{\ell s}}\bigg), \end{align} $$

where

(2.30) $$ \begin{align} A(p^{2}) &= \lambda_{f}^{2}(p^{2})\lambda_{g}^{2}(p^{2})r_{D}^{*}(p^{2})- \frac{1}{2}\lambda_{f}^{4}(p)\lambda_{g}^{4}(p)(r_{D}^{*}(p))^{2} \nonumber \\ &= \big( 1+\lambda_{f}(p^{2}) + \lambda_{f}(p^{4})\big)\big( 1+\lambda_{g}(p^{2}) + \lambda_{g}(p^{4})\big)\big(1+ \chi_{D}(p) + \chi_{D}(p^{2})\big) \nonumber \\ &\quad - \frac{1}{2}\big( 2+3\lambda_{\textrm{sym}^{2}f}(p) + \lambda_{\textrm{sym}^{4}f}(p)\big)\nonumber \\ &\quad \times\big( 2+3\lambda_{\textrm{sym}^{2}g}(p) + \lambda_{\textrm{sym}^{4}g}(p)\big)(1+2\chi_{D}(p) + \chi_{D}(p^{2}))\nonumber \\ &\ll |\alpha_{f}(p)|^{8} + |\beta_{f}(p)|^{8}, \end{align} $$

StartLayout 1st Row 1st Column upper A left parenthesis p squared right parenthesis 2nd Column equals lamda Subscript f Superscript 2 Baseline left parenthesis p squared right parenthesis lamda Subscript g Superscript 2 Baseline left parenthesis p squared right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p squared right parenthesis minus one half lamda Subscript f Superscript 4 Baseline left parenthesis p right parenthesis lamda Subscript g Superscript 4 Baseline left parenthesis p right parenthesis left parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis right parenthesis squared 2nd Row 1st Column Blank 2nd Column equals left parenthesis 1 plus lamda Subscript f Baseline left parenthesis p squared right parenthesis plus lamda Subscript f Baseline left parenthesis p Superscript 4 Baseline right parenthesis right parenthesis left parenthesis 1 plus lamda Subscript g Baseline left parenthesis p squared right parenthesis plus lamda Subscript g Baseline left parenthesis p Superscript 4 Baseline right parenthesis right parenthesis left parenthesis 1 plus chi Subscript upper D Baseline left parenthesis p right parenthesis plus chi Subscript upper D Baseline left parenthesis p squared right parenthesis right parenthesis 3rd Row 1st Column Blank 2nd Column minus one half left parenthesis 2 plus 3 lamda Subscript sym squared f Baseline left parenthesis p right parenthesis plus lamda Subscript sym Sub Superscript 4 Subscript f Baseline left parenthesis p right parenthesis right parenthesis 4th Row 1st Column Blank 2nd Column times left parenthesis 2 plus 3 lamda Subscript sym squared g Baseline left parenthesis p right parenthesis plus lamda Subscript sym Sub Superscript 4 Subscript g Baseline left parenthesis p right parenthesis right parenthesis left parenthesis 1 plus 2 chi Subscript upper D Baseline left parenthesis p right parenthesis plus chi Subscript upper D Baseline left parenthesis p squared right parenthesis right parenthesis 5th Row 1st Column Blank 2nd Column much less than StartAbsoluteValue alpha Subscript f Baseline left parenthesis p right parenthesis EndAbsoluteValue Superscript 8 Baseline plus StartAbsoluteValue beta Subscript f Baseline left parenthesis p right parenthesis EndAbsoluteValue Superscript 8 Baseline comma EndLayout

by appealing to (2.1), (2.2), (2.4), and (2.28). We have

$$ \begin{align*} \nonumber \lambda_{f}^{2}(p)\lambda_{g}^{2}(p)r_{D}^{*}(p) &= (1+ \lambda_{f}(p^{2}))(1+ \lambda_{g}(p^{2}))r_{D}^{*}(p) \nonumber \\ &=(1+\lambda_{\textrm{sym}^{2}f}(p))(1+\lambda_{\textrm{sym}^{2}g}(p))(1+\chi_{D}(p)) \nonumber \\ &= 1+ \lambda_{\textrm{sym}^{2}f}(p) + \lambda_{\textrm{sym}^{2}g}(p) + \lambda_{\textrm{sym}^{2}f\times \textrm{sym}^{2}g}(p) \nonumber \\ &+ \chi_{D}(p) + \lambda_{\textrm{sym}^{2}f\otimes\chi_{D}}(p) + \lambda_{\textrm{sym}^{2}g\otimes\chi_{D}}(p) + \lambda_{\textrm{sym}^{2}f\times \textrm{sym}^{2}g\otimes\chi_{D}}(p). \nonumber \end{align*} $$

StartLayout 1st Row 1st Column lamda Subscript f Superscript 2 Baseline left parenthesis p right parenthesis lamda Subscript g Superscript 2 Baseline left parenthesis p right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis 2nd Column equals left parenthesis 1 plus lamda Subscript f Baseline left parenthesis p squared right parenthesis right parenthesis left parenthesis 1 plus lamda Subscript g Baseline left parenthesis p squared right parenthesis right parenthesis r Subscript upper D Superscript asterisk Baseline left parenthesis p right parenthesis 2nd Row 1st Column Blank 2nd Column equals left parenthesis 1 plus lamda Subscript sym squared f Baseline left parenthesis p right parenthesis right parenthesis left parenthesis 1 plus lamda Subscript sym squared g Baseline left parenthesis p right parenthesis right parenthesis left parenthesis 1 plus chi Subscript upper D Baseline left parenthesis p right parenthesis right parenthesis 3rd Row 1st Column Blank 2nd Column equals 1 plus lamda Subscript sym squared f Baseline left parenthesis p right parenthesis plus lamda Subscript sym squared g Baseline left parenthesis p right parenthesis plus lamda Subscript sym squared f times sym squared g Baseline left parenthesis p right parenthesis 4th Row 1st Column Blank 2nd Column plus chi Subscript upper D Baseline left parenthesis p right parenthesis plus lamda Subscript sym squared f circled times chi Sub Subscript upper D Subscript Baseline left parenthesis p right parenthesis plus lamda Subscript sym squared g circled times chi Sub Subscript upper D Subscript Baseline left parenthesis p right parenthesis plus lamda Subscript sym squared f times sym squared g circled times chi Sub Subscript upper D Subscript Baseline left parenthesis p right parenthesis period EndLayout

Define

(2.31) $$ \begin{align} \mathcal{G}(s) &:= \zeta(s)L(\chi_{D}, s)L(\textrm{sym}^{2}f, s)L(\textrm{sym}^{2}g, s)L(\textrm{sym}^{2}f\times \textrm{sym}^{2}g, s)\nonumber \\ &\quad \cdot L(\textrm{sym}^{2}f\otimes\chi_{D}, s)L(\textrm{sym}^{2}g\otimes\chi_{D}, s)L(\textrm{sym}^{2}f\times \textrm{sym}^{2}g\otimes\chi_{D}, s). \end{align} $$

Then, we can deduce that the factor of $\mathfrak {G}_{1}(s)$ corresponding to the coefficient $p^{-s}$ is given by $\mathcal {G}(s)$ . Note that

(2.32) $$ \begin{align} \log \mathcal{G}(s) = \sum_{p}\bigg( \frac{\lambda_{f}^{2}(p)\lambda_{g}^{2}(p)r_{D}^{*}(p) }{p^{s}} + \frac{R(p^{2})}{p^{2s}} + \sum_{\ell\geqslant 3}\frac{R(p^{\nu})}{p^{\ell s}} \bigg). \end{align} $$

From the Euler product of $\zeta (s)$ , it follows that

(2.33) $$ \begin{align} \log\zeta(s) = \sum_{p} \bigg( \frac{1}{p^{s}} + \frac{1/2}{p^{2s}} + \cdots\bigg). \end{align} $$ log zeta left parenthesis s right parenthesis equals sigma summation Underscript p Endscripts left parenthesis StartFraction 1 Over p Superscript s Baseline EndFraction plus StartFraction 1 divided by 2 Over p Superscript 2 s Baseline EndFraction plus midline horizontal ellipsis right parenthesis period

Similarly,

(2.34) $$ \begin{align} \log L(\textrm{sym}^{2j}f, s) &= - \!\sum_{p} \!\bigg\{ \sum_{i=1}^{j} \log\bigg( 1-\frac{\alpha_{f}^{2i}(p)}{p^{s}} \!\bigg) + \log\bigg( 1-\frac{1}{p^{s}} \!\bigg)+ \sum_{\ell=1}^{j} \log\bigg( 1-\frac{\alpha_{f}^{2\ell}(p)}{p^{s}} \!\bigg)\bigg\} \nonumber \\ &= \sum_{p} \bigg( \frac{\lambda_{\mathrm{sym}^{2j}f}(p)}{p^{s}} + \frac{\rho_{j}(p^{2})}{p^{2s}} + \cdots\bigg), \end{align} $$

StartLayout 1st Row 1st Column log upper L left parenthesis sym Superscript 2 j Baseline f comma s right parenthesis 2nd Column equals minus sigma summation Underscript p Endscripts left brace sigma summation Underscript i equals 1 Overscript j Endscripts log left parenthesis 1 minus StartFraction alpha Subscript f Superscript 2 i Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction right parenthesis plus log left parenthesis 1 minus StartFraction 1 Over p Superscript s Baseline EndFraction right parenthesis plus sigma summation Underscript script l equals 1 Overscript j Endscripts log left parenthesis 1 minus StartFraction alpha Subscript f Superscript 2 script l Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction right parenthesis right brace 2nd Row 1st Column Blank 2nd Column equals sigma summation Underscript p Endscripts left parenthesis StartFraction lamda Subscript sym Sub Superscript 2 j Subscript f Baseline left parenthesis p right parenthesis Over p Superscript s Baseline EndFraction plus StartFraction rho Subscript j Baseline left parenthesis p squared right parenthesis Over p Superscript 2 s Baseline EndFraction plus midline horizontal ellipsis right parenthesis comma EndLayout

where

$$ \begin{align*} \nonumber \rho_{j}(p^{2}) = \frac{1}{2}\sum_{i=0}^{2j} (-1)^{i} \lambda_{f}(p^{2i}). \end{align*} $$ rho Subscript j Baseline left parenthesis p squared right parenthesis equals one half sigma summation Underscript i equals 0 Overscript 2 j Endscripts left parenthesis negative 1 right parenthesis Superscript i Baseline lamda Subscript f Baseline left parenthesis p Superscript 2 i Baseline right parenthesis period

One can also attain

(2.35) $$ \begin{align} \log L(\textrm{sym}^{2}f\times \textrm{sym}^{2}g, s) = \sum_{p} \bigg( \frac{\lambda_{\mathrm{sym}^{2}f\times \mathrm{sym}^{2}g}(p)}{p^{s}} + \frac{\varrho(p^{2})}{p^{2s}} + \cdots\bigg), \end{align} $$

where

$$ \begin{align*} \nonumber \varrho(p^{2}) = \big( \alpha_{f}^{4}(p) + 1+ \beta_{f}^{4}(p) \big)\big( \alpha_{g}^{4}(p) + 1+ \beta_{g}^{4}(p) \big). \end{align*} $$

Gathering together (2.31)–(2.35), it follows that

(2.36) $$ \begin{align} R(p^{2}) &= \frac{1}{2}\big( 1+ \chi_{D}(p^{2}) + (\lambda_{f}(p^{4})- \lambda_{f}(p^{2})+1) + (\lambda_{g}(p^{4})- \lambda_{g}(p^{2})+1)+ \varrho(p^{2}) \nonumber \\ & \quad +(\lambda_{f}(p^{4})- \lambda_{f}(p^{2})+1)\chi_{D}(p^{2}) +(\lambda_{g}(p^{4})- \lambda_{g}(p^{2})+1)\chi_{D}(p^{2})\nonumber \\ &\quad +\varrho(p^{2})\chi_{D}(p^{2}) \big) \nonumber \\ &\ll |\alpha_{f}(p)|^{8} + |\beta_{f}(p)|^{8}. \end{align} $$

StartLayout 1st Row 1st Column upper R left parenthesis p squared right parenthesis 2nd Column equals one half left parenthesis 1 plus chi Subscript upper D Baseline left parenthesis p squared right parenthesis plus left parenthesis lamda Subscript f Baseline left parenthesis p Superscript 4 Baseline right parenthesis minus lamda Subscript f Baseline left parenthesis p squared right parenthesis plus 1 right parenthesis plus left parenthesis lamda Subscript g Baseline left parenthesis p Superscript 4 Baseline right parenthesis minus lamda Subscript g Baseline left parenthesis p squared right parenthesis plus 1 right parenthesis plus rho left parenthesis p squared right parenthesis 2nd Row 1st Column Blank 2nd Column plus left parenthesis lamda Subscript f Baseline left parenthesis p Superscript 4 Baseline right parenthesis minus lamda Subscript f Baseline left parenthesis p squared right parenthesis plus 1 right parenthesis chi Subscript upper D Baseline left parenthesis p squared right parenthesis plus left parenthesis lamda Subscript g Baseline left parenthesis p Superscript 4 Baseline right parenthesis minus lamda Subscript g Baseline left parenthesis p squared right parenthesis plus 1 right parenthesis chi Subscript upper D Baseline left parenthesis p squared right parenthesis 3rd Row 1st Column Blank 2nd Column plus rho left parenthesis p squared right parenthesis chi Subscript upper D Baseline left parenthesis p squared right parenthesis right parenthesis 4th Row 1st Column Blank 2nd Column much less than StartAbsoluteValue alpha Subscript f Baseline left parenthesis p right parenthesis EndAbsoluteValue Superscript 8 Baseline plus StartAbsoluteValue beta Subscript f Baseline left parenthesis p right parenthesis EndAbsoluteValue Superscript 8 Baseline period EndLayout

Collecting together (2.27), (2.29),and (2.32), and noticing (2.30) and (2.36), we arrive at

$$ \begin{align*} \nonumber \mathfrak{G}(s) &= \mathcal{G}(s)\cdot w_{D}\exp\bigg\{ \sum_{p} \bigg(\frac{A(p^{2})-R(p^{2})}{p^{2s}} + \sum_{\nu\geqslant 3} \frac{A(p^{\nu})-R(p^{\nu})}{p^{\nu s}} \bigg)\bigg\}\nonumber \\ &=\mathcal{G}(s) \cdot w_{D}\prod_{p}\bigg( 1+\frac{A(p^{2})-R(p^{2})}{p^{2s}} + \cdots\bigg) \nonumber \\ &:= \mathcal{G}(s)V(s), \nonumber \end{align*} $$

StartLayout 1st Row 1st Column German upper G left parenthesis s right parenthesis 2nd Column equals script upper G left parenthesis s right parenthesis dot w Subscript upper D Baseline exp left brace sigma summation Underscript p Endscripts left parenthesis StartFraction upper A left parenthesis p squared right parenthesis minus upper R left parenthesis p squared right parenthesis Over p Superscript 2 s Baseline EndFraction plus sigma summation Underscript nu greater than or slanted equals 3 Endscripts StartFraction upper A left parenthesis p Superscript nu Baseline right parenthesis minus upper R left parenthesis p Superscript nu Baseline right parenthesis Over p Superscript nu s Baseline EndFraction right parenthesis right brace 2nd Row 1st Column Blank 2nd Column equals script upper G left parenthesis s right parenthesis dot w Subscript upper D Baseline product Underscript p Endscripts left parenthesis 1 plus StartFraction upper A left parenthesis p squared right parenthesis minus upper R left parenthesis p squared right parenthesis Over p Superscript 2 s Baseline EndFraction plus midline horizontal ellipsis right parenthesis 3rd Row 1st Column Blank 2nd Column colon equals script upper G left parenthesis s right parenthesis upper V left parenthesis s right parenthesis comma EndLayout

where for $\sigma>\frac {1}{2}$ ,

$$ \begin{align*} \nonumber V(s) \ll \prod_{p} \bigg( 1+\frac{|\alpha_{f}(p)|^{8} + |\beta_{f}(p)|^{8}}{p^{2\sigma}}\bigg) \ll 1, \end{align*} $$

by noticing the upper bound (2.25). Hence, $V(s)$ denotes an absolutely convergent Dirichlet series for $\Re (s)>\frac {1}{2}$ .

We conclude this section by providing a general criterion to detect sign changes of arithmetic functions attributed to Lao and Qiao [Reference Lao and Qiao37, Lemma 3.7], which is built upon the recent work of Lowry-Duda [Reference Lowry-Duda39].

Lemma 2.7 [Reference Lao and Qiao37, Lemma 3.7]

Let $\xi (n)\geqslant 0$ be any fixed weight, and let $\alpha ,\beta ,\gamma ,$ and $\eta $ be positive constants. Suppose that the real sequence $(c(n))$ satisfies the following axioms:

(i) For any $\varepsilon>0$ , we have $c(n)= O(n^{\alpha +\varepsilon })$ .

(ii) We have

$$ \begin{align*} \sum_{n\leqslant x} c(n)\xi(n) = O(x^{\beta+\varepsilon}). \end{align*} $$

(iii) We have

$$ \begin{align*} \sum_{n\leqslant x} c^{2}(n)\xi(n) = x^{\gamma}P_{j}(\log x) + O(x^{\eta+\varepsilon}). \end{align*} $$

Here, $P_{j}(t)$ denote a polynomial in t of degree j.

Then, for any $\delta $ satisfying the inequality,

$$ \begin{align*} \max(\alpha+\beta,\eta)-(\gamma-1)< \delta< 1, \end{align*} $$

the sequence $(c(n))$ has at least one sign change in the interval $(x,x + x^{\delta }]$ , for sufficiently large $x>0$ .

3 The main propositions

In this section, we are dedicated to the estimations of $S(x; f,g, D)$ and $T(x;f,g, D)$ asymptotically, defined in (2.7) and (2.8), respectively. The first main result is the following.

Proposition 3.1 Let $S(x; f,g, D)$ be defined as (2.7). For sufficiently large $x>0$ , we have

$$ \begin{align*} S(x; f,g, D) \ll x^{\frac{3}{4}+\varepsilon}, \end{align*} $$

for any $\varepsilon>0$ .

Proof To prove the asymptotic upper bound for $S(x; f,g, D)$ , we employ a general result given in [Reference Hua15, Lemma 2.3]. From the discussion of [Reference Lü and Ma40, Remark 1.5], we learn that

$$ \begin{align*} \sum_{n\leqslant x}|\lambda_{\pi\times\pi'}(n)|^{2} \ll x(\log x)^{4} \end{align*} $$

with $\pi $ and $\pi '$ being self-contragredient automorphic cuspidal representations on $GL_{2}(\mathbb {A}_{\mathbb {Q}})$ with unitary central character, which in turns implies that, for any $\varepsilon>0$ ,

$$ \begin{align*} \sum_{n\leqslant x}|\lambda_{f\times g}(n)|^{2} \ll x^{1+\varepsilon}, \qquad \sum_{n\leqslant x}|\lambda_{f\times g\otimes\chi_{D}}(n)|^{2} \ll x^{1+\varepsilon}. \end{align*} $$

Furthermore, for $\mathfrak {F}(s)$ as defined by (2.9), it can be verified that all the other conditions in [Reference Hua15, Lemma 2.3] (see also [Reference Jiang and Lü25, Corollary 1.4]) are plainly satisfied. Henceforth, on applying [Reference Hua15, Lemma 2.3] to Lemma 2.5 with $\sigma _{0}=\frac {1}{2}+\varepsilon $ and $d=8$ , one can deduce that

(3.1) $$ \begin{align} S(x; f,g, D) = O\big( x^{\frac{3}{2}-\kappa+\varepsilon}\big) + O\big( \widetilde{E}(x)\big) \end{align} $$

for any $0< \kappa < 1$ , where $\widetilde {E}(x)$ is given by

$$ \begin{align*} \nonumber \widetilde{E}(x):= x^{\kappa\varepsilon}\sum_{x< n< x+x^{\kappa}} \big| \lambda_{f}(n)\lambda_{g}(n)\big|, \end{align*} $$

by virtue of the bound $r_{D}(n) \ll n^{\varepsilon }$ . Note that $Q_{\mathscr {B}}$ in [Reference Hua15, Lemma 2.3] is the arithmetic conductor, which is not necessarily concerned in the context. Here, we employed the fact that $L(f\times g, s)$ and $L(f\times g\otimes \chi ,s)$ are entire functions, as long as f and g are distinct cuspidal Hecke–Maass forms.

Now, it is crucial to estimate $\widetilde {E}(x)$ appropriately. Using the estimate originated from Rankin–Selberg theory (cf. [Reference Jiang, Lü and Wang26, p. 379]), coupled with a special case of Ramakrishnan’s modularity theorem [Reference Ramakrishnan46] which asserts that there exists a cuspidal representation $\Pi =\pi _{f\times g}$ on $GL_{4}(\mathbb {A}_{\mathbb {Q}})$ such that

$$ \begin{align*} L(f\times g, s) = L(\Pi,s), \end{align*} $$

we learn that

$$ \begin{align*} \sum_{n\leqslant x}\lambda_{f\times g}^{2}(n) = (c_{f\times g}+o(1)) x,\qquad x\to\infty, \end{align*} $$

where $c_{f\times g}>0$ is some suitable constant. By adapting the argument of Lemma $2.6$ , one can also derive that

(3.2) $$ \begin{align} \sum_{n=1}^{\infty}\frac{\lambda_{f}^{2}(n)\lambda_{g}^{2}(n)}{n^{s}} = \sum_{n=1}^{\infty}\frac{\lambda_{f\times g}^{2}(n)}{n^{s}}\mathscr{V}(s), \end{align} $$

where $\mathscr {V}(s)=\sum _{\nu \geqslant 1}b(\nu )\nu ^{-s}$ is an absolutely convergent Dirichlet series for $\Re (s)>\frac {1}{2}$ . From (3.2), we know that

$$ \begin{align*} \lambda_{f}^{2}(n)\lambda_{g}^{2}(n) = \sum_{n=u\nu}\lambda_{f\times g}^{2}(u)b(\nu), \end{align*} $$

satisfying the constraint

$$ \begin{align*} \sum_{\nu\geqslant 1}\frac{|b(\nu)|}{\nu^{\sigma}} \ll_{\sigma} 1, \qquad \sigma>\frac{1}{2}. \end{align*} $$ sigma summation Underscript nu greater than or slanted equals 1 Endscripts StartFraction StartAbsoluteValue b left parenthesis nu right parenthesis EndAbsoluteValue Over nu Superscript sigma Baseline EndFraction much less than Subscript sigma Baseline 1 comma sigma greater than one half period

Therefore, we deduce that

(3.3) $$ \begin{align} \sum_{n\leqslant x} \lambda_{f}^{2}(n)\lambda_{g}^{2}(n) &= \sum_{\nu\leqslant x} b(\nu)\sum_{u\leqslant x/\nu}\lambda_{f\times g}^{2}(u) \nonumber \\ &= \sum_{\nu\leqslant x} b(\nu) (c_{f\times g}+o(1) )\frac{x}{\nu} \nonumber \\ &= (c_{f\times g}^{*}+o(1)) x,\qquad x\to\infty, \end{align} $$

StartLayout 1st Row 1st Column sigma summation Underscript n less than or slanted equals x Endscripts lamda Subscript f Superscript 2 Baseline left parenthesis n right parenthesis lamda Subscript g Superscript 2 Baseline left parenthesis n right parenthesis 2nd Column equals sigma summation Underscript nu less than or slanted equals x Endscripts b left parenthesis nu right parenthesis sigma summation Underscript u less than or slanted equals x divided by nu Endscripts lamda Subscript f times g Superscript 2 Baseline left parenthesis u right parenthesis 2nd Row 1st Column Blank 2nd Column equals sigma summation Underscript nu less than or slanted equals x Endscripts b left parenthesis nu right parenthesis left parenthesis c Subscript f times g Baseline plus o left parenthesis 1 right parenthesis right parenthesis StartFraction x Over nu EndFraction 3rd Row 1st Column Blank 2nd Column equals left parenthesis c Subscript f times g Superscript asterisk Baseline plus o left parenthesis 1 right parenthesis right parenthesis x comma x right arrow infinity comma EndLayout

where $c_{f\times g}^{*}>0$ is a certain suitable constant, and $b(\nu )$ ’s are the coefficients of $\mathscr {V}(s)$ . Henceforth, by the Cauchy–Schwarz inequality and the estimate (3.3), one can obtain

(3.4) $$ \begin{align} \widetilde{E}(x) \ll x^{\frac{\kappa}{2}+\varepsilon}\bigg( \sum_{x<n <x+x^{\kappa}}\lambda_{f}^{2}(n)\lambda_{g}^{2}(n) \bigg)^{\frac{1}{2}} \ll x^{\kappa+\varepsilon}. \end{align} $$

By substituting (3.4) back into (3.1), we obtain

(3.5) $$ \begin{align} S(x; f,g, D) = O\big( x^{\frac{3}{2}-\kappa+\varepsilon}\big) + O\big( x^{\kappa+\varepsilon}\big). \end{align} $$

By choosing $\kappa =\frac {3}{4}$ in (3.5), we eventually conclude that

$$ \begin{align*} S(x; f,g, D) \ll x^{\frac{3}{4}+\varepsilon} \end{align*} $$

for any $\varepsilon>0$ . This completes the proof of Proposition $3.1$ .

In the next result, we come to the estimation of $T(x;f,g, D)$ .

Proposition 3.2 Let $T(x; f,g, D)$ be defined as (2.8). For sufficiently large $x>0$ and any $\varepsilon>0$ , we have

$$ \begin{align*} \nonumber T(x; f,g, D) = C_{f,g,D}x + O\big( x^{\frac{277}{298}+\varepsilon}\big), \end{align*} $$

where $C_{f,g,D}>0$ is a constant depending on $f,g,$ and D.

Proof Let $1\leqslant y< \frac {x}{2}$ . We introduce a smooth compactly supported function $w(t)$ conforming to the following constrains:

(i) $w(t)=1$ for $t\in [2y,x]$ ;

(ii) $w(t)=0$ for $t<y$ and $t>x+y$ ;

(iii) assume that $w^{(r)}(t)\ll _{r} y^{-r}$ for all integers $r\geqslant 0$ , where $w^{(r)}(t)$ is the r-th-order derivative of $w(t)$ .

For simplicity, we write $\mathfrak {f}(n) := \lambda _{f}^{2}(n)\lambda _{g}^{2}(n)r_{D}(n)$ . Following [Reference Jiang and Lü25, Section 3], we have

(3.6) $$ \begin{align} \sum_{n\leqslant x} \mathfrak{f}(n)= \sum_{n=1}^{\infty} \mathfrak{f}(n)w(n) + O\bigg( \sum_{n<2y} |\mathfrak{f}(n)|\bigg) + O\bigg( \sum_{x<n<x+y}|\mathfrak{f}(n)|\bigg). \end{align} $$

By employing Mellin’s inverse transform, it follows that

(3.7) $$ \begin{align} \sum_{n=1}^{\infty}\mathfrak{f}(n)w(n) = \frac{1}{2\pi i} \int_{(b)} \widetilde{w}(s) \mathfrak{G}(s)ds, \end{align} $$

where $\widetilde {w}(s)$ is the Mellin transform of w given by

$$ \begin{align*} \widetilde{w}(s) = \int_{0}^{\infty} w(t) t^{s}\frac{dt}{t}, \end{align*} $$

and b is a real number larger than the abscissa of the absolute convergence of the Dirichlet series $\mathcal {F}(s)$ . Furthermore, by integration by parts, we observe

(3.8) $$ \begin{align} \widetilde{w}(s) =\frac{1}{s(s+1)\dots(s+m-1)}\int_{0}^{\infty} w^{(m)}(t) t^{s+m-1} dt \ll \frac{y}{x^{1-\sigma}}\bigg( \frac{x}{|s|y}\bigg)^{m}, \end{align} $$

for any $m\geqslant 1$ , where $\sigma =\Re (s)$ . By shifting the line of integration from $\Re (s)=1+\varepsilon $ to the line $\Re (s)=\frac {1}{2}+\varepsilon $ , in combination with Cauchy’s residue theorem, one can obtain

(3.9) $$ \begin{align} \sum_{n=1}^{\infty}\mathfrak{f}(n)w(n) &= \frac{1}{2\pi i} \int_{(1+\varepsilon)} \widetilde{w}(s) \mathfrak{G}(s)ds \nonumber \\ &= \mathop{\textrm{Res}}_{s=1}\bigg( \frac{x^{s}}{s}\mathfrak{G}(s)\bigg) + \frac{1}{2\pi i} \int_{(\frac{1}{2}+\varepsilon)} \widetilde{w}(s) \mathfrak{G}(s) ds. \end{align} $$

StartLayout 1st Row 1st Column sigma summation Underscript n equals 1 Overscript infinity Endscripts German f left parenthesis n right parenthesis w left parenthesis n right parenthesis 2nd Column equals StartFraction 1 Over 2 pi i EndFraction integral Underscript left parenthesis 1 plus epsilon right parenthesis Endscripts ModifyingAbove w With tilde left parenthesis s right parenthesis German upper G left parenthesis s right parenthesis d s 2nd Row 1st Column Blank 2nd Column equals Res Underscript s equals 1 Endscripts left parenthesis StartFraction x Superscript s Baseline Over s EndFraction German upper G left parenthesis s right parenthesis right parenthesis plus StartFraction 1 Over 2 pi i EndFraction integral Underscript left parenthesis one half plus epsilon right parenthesis Endscripts ModifyingAbove w With tilde left parenthesis s right parenthesis German upper G left parenthesis s right parenthesis d s period EndLayout

Now let $T=: \frac {x^{1+\varepsilon }}{y}$ with y being determined later as a power of x. By taking sufficiently large $m>0$ in (3.8), we may derive that

(3.10) $$ \begin{align} \frac{1}{2\pi i} \int_{(\frac{1}{2}+\varepsilon)} \widetilde{w}(s) \mathfrak{G}(s) ds = \frac{1}{2\pi i} \int_{\frac{1}{2}+\varepsilon-iT}^{\frac{1}{2}+\varepsilon+iT} \widetilde{w}(s) \mathfrak{G}(s) ds + O\big( x^{-A}\big), \end{align} $$

where $A>0$ is a sufficiently large positive number. Additionally, by appealing to (3.3) and taking into consideration of the bound $r_{D}(n) \ll n^{\varepsilon }$ , one can attain

(3.11) $$ \begin{align} O\bigg( \sum_{n<2y} |\mathfrak{f}(n)|\bigg) + O\bigg( \sum_{x<n<x+y}|\mathfrak{f}(n)|\bigg) \ll y^{1+\varepsilon}. \end{align} $$

Henceforth, by assembling the estimates (3.6), (3.10), and (3.11), we may deduce that

(3.12) $$ \begin{align} \sum_{n\leqslant x} \mathfrak{f}(n) = \mathop{\textrm{Res}}_{s=1}\bigg( \frac{x^{s}}{s}\mathfrak{G}(s)\bigg) + E + O\big( x^{-A}\big) + O\big( y^{1+\varepsilon}\big), \end{align} $$

where $A>0$ is sufficiently large, and

(3.13) $$ \begin{align} E := \frac{1}{2\pi i} \int_{\frac{1}{2}+\varepsilon-iT}^{\frac{1}{2}+\varepsilon+iT}\widetilde{w}(s) \mathfrak{G}(s) ds. \end{align} $$ upper E colon equals StartFraction 1 Over 2 pi i EndFraction integral Subscript one half plus epsilon minus i upper T Superscript one half plus epsilon plus i upper T Baseline ModifyingAbove w With tilde left parenthesis s right parenthesis German upper G left parenthesis s right parenthesis d s period

To obtain the asymptotic formula for the left-hand side of (3.12), it is sufficient to estimate E and the residue of the associated L-series at $s=1$ appropriately.

From Lemma $2.6$ , we know that $\mathfrak {G}(s)$ has a simple pole at $s=1$ , which comes from the factor $\zeta (s)$ . Henceforth,

(3.14) $$ \begin{align} \mathop{\textrm{Res}}_{s=1}\bigg( \frac{x^{s}}{s}\mathfrak{G}(s)\bigg) &= L(1,\chi_{D})L(\textrm{sym}^{2}f, 1)L(\textrm{sym}^{2}g, 1)\cdot L(\textrm{sym}^{2}f\times \textrm{sym}^{2}g, 1)\nonumber \\ &\quad L(\textrm{sym}^{2}f\otimes\chi_{D}, 1)\cdot L(\textrm{sym}^{2}g\otimes\chi_{D}, 1)\nonumber \\ &\quad L(\textrm{sym}^{2}f\times \textrm{sym}^{2}g\otimes\chi_{D}, 1)V(1)x :=C_{f,g,D}x, \end{align} $$

StartLayout 1st Row 1st Column Res Underscript s equals 1 Endscripts left parenthesis StartFraction x Superscript s Baseline Over s EndFraction German upper G left parenthesis s right parenthesis right parenthesis 2nd Column equals upper L left parenthesis 1 comma chi Subscript upper D Baseline right parenthesis upper L left parenthesis sym squared f comma 1 right parenthesis upper L left parenthesis sym squared g comma 1 right parenthesis dot upper L left parenthesis sym squared f times sym squared g comma 1 right parenthesis 2nd Row 1st Column Blank 2nd Column upper L left parenthesis sym squared f circled times chi Subscript upper D Baseline comma 1 right parenthesis dot upper L left parenthesis sym squared g circled times chi Subscript upper D Baseline comma 1 right parenthesis 3rd Row 1st Column Blank 2nd Column upper L left parenthesis sym squared f times sym squared g circled times chi Subscript upper D Baseline comma 1 right parenthesis upper V left parenthesis 1 right parenthesis x colon equals upper C Subscript f comma g comma upper D Baseline x comma EndLayout

which produces the main term for the expression $T(x; f,g, D)$ . Here, $C_{f,g,D}>0$ is the constant, given in (3.14), relying on $f,g,$ and D.

Now, we are left to deal with (3.13). Let $\mathcal {G}(s)$ be defined as (2.31). Applying (3.8) with $m=1$ , noting $V(s)\ll 1$ in the region $\Re (s)\geqslant \frac {1}{2}+\varepsilon $ , we can derive that

$$ \begin{align*} E &= \frac{1}{2\pi i} \int_{\frac{1}{2}+\varepsilon-iT}^{\frac{1}{2}+\varepsilon+iT}\widetilde{w}(s) \mathfrak{G}(s) ds\nonumber \\ &\ll x^{\frac{1}{2}+\varepsilon}\int_{6}^{T}\frac{\mathcal{G}(\frac{1}{2}+\varepsilon+it)}{|\frac{1}{2}+\varepsilon+it|}dt + x^{\frac{1}{2}+\varepsilon}\int_{0}^{6} \frac{\mathcal{G}(\frac{1}{2}+\varepsilon+it)}{|\frac{1}{2}+\varepsilon+it|}dt \nonumber \\ &\ll x^{\frac{1}{2}+\varepsilon}\int_{1}^{T}\frac{\mathcal{G}(\frac{1}{2}+\varepsilon+it)}{t}dt +x^{\frac{1}{2}+\varepsilon}. \nonumber \end{align*} $$

StartLayout 1st Row 1st Column upper E 2nd Column equals StartFraction 1 Over 2 pi i EndFraction integral Subscript one half plus epsilon minus i upper T Superscript one half plus epsilon plus i upper T Baseline ModifyingAbove w With tilde left parenthesis s right parenthesis German upper G left parenthesis s right parenthesis d s 2nd Row 1st Column Blank 2nd Column much less than x Superscript one half plus epsilon Baseline integral Subscript 6 Superscript upper T Baseline StartFraction script upper G left parenthesis one half plus epsilon plus i t right parenthesis Over StartAbsoluteValue one half plus epsilon plus i t EndAbsoluteValue EndFraction d t plus x Superscript one half plus epsilon Baseline integral Subscript 0 Superscript 6 Baseline StartFraction script upper G left parenthesis one half plus epsilon plus i t right parenthesis Over StartAbsoluteValue one half plus epsilon plus i t EndAbsoluteValue EndFraction d t 3rd Row 1st Column Blank 2nd Column much less than x Superscript one half plus epsilon Baseline integral Subscript 1 Superscript upper T Baseline StartFraction script upper G left parenthesis one half plus epsilon plus i t right parenthesis Over t EndFraction d t plus x Superscript one half plus epsilon Baseline period EndLayout

By the Cauchy–Schwarz inequality, and invoking Lemmas $2.1$ – $2.4$ , together with $T= \frac {x^{1+\varepsilon }}{y}$ and $1\leqslant y< \frac {x}{2}$ , we derive that

(3.15) $$ \begin{align} E &\ll x^{\frac{1}{2}+\varepsilon}\log T\sup_{1\leqslant T_{1}\leqslant T}T_{1}^{-1} \bigg| \zeta\bigg( \frac{1}{2}+iT_{1} \bigg) L\bigg( \frac{1}{2}+iT_{1},\chi_{D} \bigg)L\bigg( \textrm{sym}^{2}f, \frac{1}{2}+iT_{1} \bigg)\nonumber \\ &\quad \cdot L\bigg( \textrm{sym}^{2}g, \frac{1}{2}+iT_{1} \bigg)L\bigg( \textrm{sym}^{2}f\otimes\chi_{D}, \frac{1}{2}+iT_{1} \bigg)L\bigg( \textrm{sym}^{2}g\otimes\chi_{D}, \frac{1}{2}+iT_{1} \bigg)\bigg| \nonumber \\ &\quad \cdot \bigg( \int_{T_{1}}^{2T_{1}}\bigg| L\bigg( \textrm{sym}^{2}f\times\textrm{sym}^{2}g, \frac{1}{2}+it \bigg)\bigg|^{2} dt \bigg)^{\frac{1}{2}}\nonumber \\ &\quad \cdot \bigg( \int_{T_{1}}^{2T_{1}}\bigg| L\bigg( \textrm{sym}^{2}f\times\textrm{sym}^{2}g\otimes\chi_{D}, \frac{1}{2}+it \bigg)\bigg|^{2} dt \bigg)^{\frac{1}{2}} + x^{\frac{1}{2}+\varepsilon}\nonumber \\ &\ll x^{\frac{1}{2}+\varepsilon}T^{(\frac{13}{42} + \frac{8}{7}\cdot2 )\cdot\frac{1}{2}\cdot 2 + 9\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot 2-1+\varepsilon} + x^{\frac{1}{2}+\varepsilon} \nonumber \\ &\ll x^{\frac{1}{2}+\varepsilon}T^{\frac{128}{21}+\varepsilon}. \end{align} $$

StartLayout 1st Row 1st Column upper E 2nd Column much less than x Superscript one half plus epsilon Baseline log upper T sup Underscript 1 less than or slanted equals upper T 1 less than or slanted equals upper T Endscripts upper T 1 Superscript negative 1 Baseline vertical bar zeta left parenthesis one half plus i upper T 1 right parenthesis upper L left parenthesis one half plus i upper T 1 comma chi Subscript upper D Baseline right parenthesis upper L left parenthesis sym squared f comma one half plus i upper T 1 right parenthesis 2nd Row 1st Column Blank 2nd Column dot upper L left parenthesis sym squared g comma one half plus i upper T 1 right parenthesis upper L left parenthesis sym squared f circled times chi Subscript upper D Baseline comma one half plus i upper T 1 right parenthesis upper L left parenthesis sym squared g circled times chi Subscript upper D Baseline comma one half plus i upper T 1 right parenthesis vertical bar 3rd Row 1st Column Blank 2nd Column dot left parenthesis integral Subscript upper T 1 Superscript 2 upper T 1 Baseline StartAbsoluteValue upper L left parenthesis sym squared f times sym squared g comma one half plus i t right parenthesis EndAbsoluteValue squared d t right parenthesis Superscript one half 4th Row 1st Column Blank 2nd Column dot left parenthesis integral Subscript upper T 1 Superscript 2 upper T 1 Baseline StartAbsoluteValue upper L left parenthesis sym squared f times sym squared g circled times chi Subscript upper D Baseline comma one half plus i t right parenthesis EndAbsoluteValue squared d t right parenthesis Superscript one half plus x Superscript one half plus epsilon 5th Row 1st Column Blank 2nd Column much less than x Superscript one half plus epsilon Baseline upper T Superscript left parenthesis StartFraction 13 Over 42 EndFraction plus eight sevenths dot 2 right parenthesis dot one half dot 2 plus 9 dot one half dot one half dot 2 minus 1 plus epsilon Baseline plus x Superscript one half plus epsilon Baseline 6th Row 1st Column Blank 2nd Column much less than x Superscript one half plus epsilon Baseline upper T Superscript StartFraction 128 Over 21 EndFraction plus epsilon Baseline period EndLayout

By substituting (3.14) and (3.15) back into (3.12), we obtain

(3.16) $$ \begin{align} T(x; f,g, D) = C_{f,g,D}x + O\big(x^{\frac{1}{2}+\varepsilon}T^{\frac{128}{21}+\varepsilon} \big) + O\big( y^{1+\varepsilon}\big). \end{align} $$

Recalling $T=\frac {x^{1+\varepsilon }}{y}$ and choosing $y= x^{\frac {277}{298}}$ , we conclude that

$$ \begin{align*} T(x; f,g, D) = C_{f,g,D}x + O\big( x^{\frac{277}{298}+\varepsilon}\big). \end{align*} $$

This completes the proof.

4 Proof of Theorem 1.1

Now, we apply Lemma $2.7$ to prove Theorem 1.1. From Propositions 3.1 and 3.2, and appealing to Lemma 2.7 with $c(n)=\lambda _{f}(n)\lambda _{g}(n)$ and $\xi (n)=r_{D}(n)$ , one can readily derive that $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n=Q(\boldsymbol {x})}$ has at least $\gg x^{1-\vartheta }$ sign changes in the interval $(x,2x]$ for sufficiently large x, where $\vartheta $ satisfies

$$ \begin{align*} \max\bigg(\frac{7}{32}+\frac{3}{4}, \frac{277}{298}\bigg) < \vartheta< 1. \end{align*} $$ max left parenthesis seven thirty seconds plus three fourths comma StartFraction 277 Over 298 EndFraction right parenthesis less than theta less than 1 period

Henceforth, the number of sign changes of the sequence $\{\lambda _{f}(n)\lambda _{g}(n)\}$ in the interval $(x,2x]$ is at least $\gg x^{1-\vartheta }$ , with $\vartheta $ in the exponent given by $\frac {31}{32} < \vartheta < 1$ .

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Competing interests

The author declares that there is no competing interests regarding the publication of this article.

Acknowledgements

The author would like to express his gratitude to Professors Guangshi Lü, Wenpeng Zhang, Bin Chen, Deyu Zhang, Huafeng Liu, Bingrong Huang, and Yujiao Jiang, Research Fellow Zhiwei Wang, and Dr. Fei Hou, Jiafan Zhang, Wei Zhang, and Zikang Dong, for their constant encouragement and valuable suggestions. The author is extremely grateful to the anonymous reviewers for careful reading of the manuscript with great patience and their professional spirit during the period of review, and also for their meticulous checking, for thoroughly reporting countless typos and inaccuracies as well as for their valuable comments. These detailed and valuable advice given by the reviewers immensely improved the quality of the manuscript and also have made the manuscript clearer and more readable.

Footnotes

This work was financially supported in part by the National Natural Science Foundation of China (Grant No. 12401011), the National Key Research and Development Program of China (Grant No. 2021YFA1000700), the Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 25JSQ054), the Science and Technology Program for Youth New Star of Shaanxi Province (Grant No. 2025ZC-KJXX-29), the Natural Science Basic Research Program of Shaanxi Province (Grant No. 2025JC-YBQN-091), and the Scientific Research Foundation for Young Talents of WNU (Grant No. 2024XJ-QNRC-01)

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Article contents

On the quantitative simultaneous sign changes of Fourier coefficients attached to Hecke–Maass forms over certain quadratic forms

Abstract

Keywords

MSC classification

Information

1 Introduction

2 Preliminaries

Lemma 2.7 [Reference Lao and Qiao37, Lemma 3.7]

3 The main propositions

4 Proof of Theorem 1.1

Data availability statement

Competing interests

Acknowledgements

Footnotes

References

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