Hecke-Maass Cusp Form Research Articles

Shifted convolution sums for GL(3)×GL(2)$GL(3)\times GL(2)$ averaged over weighted sets

AbstractLet be the Fourier coefficients of an Hecke–Maass cusp form and be those of an Hecke holomorphic or Hecke–Maass cusp form . Let and be a sequence. We show that if for some , for any , and a similar bound holds when . This improves Sun's bound and generalizes it to an average with arbitrary weights . Moreover, we demonstrate how one can recover the factorizable moduli structure given by Jutila's variant of the circle method via studying a shifted sum with weighted average. This allows us to recover Munshi's bound on the shifted sum with a fixed shift without using Jutila's circle method.

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for [Formula: see text]. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for [Formula: see text]. Furthermore, we present a conditional result regarding sign changes of these coefficients.

Hybrid subconvexity bounds for twists of GL(3) L-functions

Let [Formula: see text] be a Hecke–Maass cusp form on [Formula: see text] and [Formula: see text] a primitive Dirichlet character of prime power conductor [Formula: see text] with [Formula: see text] prime. In this paper, we will prove the following subconvexity bound [Formula: see text] for any [Formula: see text] and [Formula: see text].

GL(2) Weyl bound via a multiplicative character delta method

We use a trivial delta method with multiplicative characters for congruence detection to prove the Weyl bound for GL(2) in the [Formula: see text]-aspect for a holomorphic or Hecke–Maass cusp form of arbitrary level and nebentypus. This parallels the work of Aggarwal [2] in 2018, with the difference being the multiplicative character has a more natural connection to the twisted [Formula: see text]-function. This provides another viewpoint to understand and explore the trivial and other delta methods.

The cubic moment of Hecke–Maass cusp forms and moments of L-functions

The cubic moment of Hecke–Maass cusp forms and moments of L-functions

On the Rankin-Selberg problem in families

In this paper, we investigate the Rankin-Selberg problem over short intervals in families of holomorphic modular forms and Hecke-Maass cusp forms. We assume a Lindelöf-on-average bound for holomorphic modular forms. On the other hand, we make no assumptions for Hecke-Maass cusp forms.

On squares of Fourier coefficients twist exponential functions with applications

Let f be a Hecke–Maass cusp form of weight zero for [Formula: see text] and [Formula: see text] be the nth Fourier coefficient. For almost all [Formula: see text] we have [Formula: see text] which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than [Formula: see text] For all [Formula: see text] of type [Formula: see text] we also show that [Formula: see text] where [Formula: see text] This result relies heavily on a generalized double sum (see Theorem 1.3).

On some estimates involving Fourier coefficients of Maass cusp forms

Let [Formula: see text] be a Hecke–Maass cusp form for [Formula: see text] with Laplace eigenvalue [Formula: see text] and let [Formula: see text] be its [Formula: see text]th normalized Fourier coefficient. It is proved that, uniformly in [Formula: see text], [Formula: see text] where the implied constant depends only on [Formula: see text]. We also consider the summation function of [Formula: see text] and under the Ramanujan conjecture we are able to prove [Formula: see text] with the implied constant depending only on [Formula: see text].

Hybrid subconvexity and the partition function

We give an upper bound for the error term in the Hardy-Ramanujan-Rademacher formula for the partition function. The main input is a new hybrid subconvexity bound for the central value L(12,f×(q⋅)) in the q and spectral parameter aspects, where f is a Hecke-Maass cusp form for Γ0(N) and q is a fundamental discriminant.

The quantitative distribution of Hecke eigenvalues of Maass forms

Let f be a normalized Hecke–Maass cusp form of weight zero for the group \(SL_2({\mathbb {Z}})\). This article presents several quantitative results about the distribution of Hecke eigenvalues of f. Applications to the \(\Omega _{\pm }\)-results for the Hecke eigenvalues of f and its symmetric square sym\(^2(f)\) are also given.

On Selberg’s limit theorem for L-functions over a family of GL(n) Hecke–Maass cusp forms

On Selberg’s limit theorem for L-functions over a family of GL(n) Hecke–Maass cusp forms

Subconvexity for GL(3)×GL(2) twists

Let π be a SL(3,Z) Hecke-Maass cusp form, f be a SL(2,Z) holomorphic cusp form or Maass cusp form and χ be any non-trivial character modp, where p is prime. We show that the L-function associated with this triplet satisfyL(12,π×f×χ)≪π,f,ϵp32−116+ϵ. The method also yields the subconvex boundL(12,π⊗χ)≪π,ϵp34−132+ϵ.

Determination of GL(3) Cusp Forms by Central Values of Quadratic Twisted L-Functions

Abstract Let $\phi $ and $\phi ^{\prime}$ be two $\operatorname {GL}(3)$ Hecke–Maass cusp forms. In this paper, we prove that $\phi =\phi ^{\prime} \textrm {or }\widetilde {\phi ^{\prime}}$ if there exists a nonzero constant $\kappa $ such that $L(\frac {1}{2},\phi \otimes \chi _{8d})=\kappa L(\frac {1}{2},\phi ^{\prime}\otimes \chi _{8d})$ for all positive odd square-free positive $d$. Here, $\widetilde {\phi ^{\prime}}$ is dual form of $\phi ^{\prime}$ and $\chi _{8d}$ is the quadratic character $(\frac {8d}{\cdot })$. To prove this, we obtain asymptotic formulas for twisted 1st moment of central values of quadratic twisted $L$-functions on $\operatorname {GL}(3)$, which will have many other applications.

Quantitative non-vanishing of central values of certain L-functions on $${{\text {GL}}}(2)\times {{\text {GL}}}(3)$$

Let $\phi$ be an even Hecke-Maass cusp form on ${\rm SL}_2(\mathbb{Z})$ whose $L$-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of $\phi$, $f$ and $\bar f$, where $f$ runs over an orthonormal basis $H_k$ of Hecke eigen elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$ of a fixed weight $k\geq 4$. As an application, we prove a quantitative non-vanishing results on the central values for the family of degree $6$ $L$-functions $L(s,\phi \times {\rm Ad}\,f)$ with $f$ in the union of $H_k$ $({\rm K} \leq k < 2{\rm K})$ as ${\rm K}\rightarrow \infty$.

An improved spectral large sieve inequality for ${\rm SL}_3(\mathbb {Z})$

We prove an improved spectral large sieve inequality for the family of ${\rm SL }_3(\mathbb {Z})$ Hecke–Maass cusp forms. The method of proof uses duality, and its structure reveals unexpected connections to Heath-Brown’s large sieve for cubic characters.

On the exceptional set of the generalized Ramanujan conjecture for GL(3)

Recently Luo and Zhou showed that for a fixed GL(2) Hecke-Maass cusp form, the natural density of primes at which the Satake parameters fail the Ramanujan Conjecture does not exceed 1/35. In this short note, we investigate the GL(3) case and obtain two similar (conditional) results.

A $$\mathrm {GL}_3$$ analog of Selberg’s result on S(t)

Let $$S(t,F):=\pi ^{-1}\arg L\big (\frac{1}{2}+it,F\big ),$$ where F is a Hecke–Maass cusp form for $$\mathrm {SL}_3({\mathbb {Z}}).$$ We establish an asymptotic formula for the spectral moments of S(t, F), and obtain several other results on S(t, F).

The first simultaneous sign change for Fourier coefficients of Hecke–Maass forms

Let $f$ and $g$ be two Hecke-Maass cusp forms of weight zero for $SL_2(\mathbb Z)$ with Laplacian eigenvalues $\frac{1}{4}+u^2$ and $\frac{1}{4}+v^2$, respectively. Then both have real Fourier coefficients say, $\lambda_f(n)$ and $\lambda_g(n)$, and we may normalize $f$ and $g$ so that $\lambda_f(1)=1=\lambda_g(1)$. In this article, we first prove that the sequence $\{\lambda_f(n)\lambda_g(n)\}_{n \in \mathbb{N}}$ has infinitely many sign changes. Then we derive a bound for the first negative coefficient for the same sequence in terms of the Laplacian eigenvalues of $f$ and $g$.

An exponential sum involving Fourier coefficients of eigenforms for $$SL(2,\pmb {{\mathbb {Z}}})$$

Let $$\lambda _f (n)$$ denote the normalized n-th Fourier coefficient of a holomorphic Hecke eigencuspform or a Hecke–Maass cusp form for the full modular group. In this paper we shall exhibit cancellations in the following sum: $$\begin{aligned} \sum _{N<n \leqslant 2N} \lambda _f(n) \nu (n) e\left( \alpha n^{\theta }\right) , \end{aligned}$$ where $$\alpha , \ \theta $$ are real numbers with $$0< \theta <1$$ , and $$\nu (n)$$ is either $$\mu (n)$$ or $$\Lambda (n)$$ .

On the sup-norm of SL 3 Hecke–Maass cusp forms

This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.