Related papers: On the Rankin--Selberg problem

In the present paper we prove lower bounds for L-functions from the Selberg class, by this means improving earlier results obtained by the second author together with J\"orn Steuding. We formulate two theorems which use slightly different…

Let $\pi$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_n$ over a number field, and let $\tilde{\pi}$ be contragredient to $\pi$. We prove effective upper and lower bounds of the correct order in the short interval prime…

This paper studies the Fourier expansion of Hecke-Maass eigenforms for $GL(2, \mathbb Q)$ of arbitrary weight, level, and character at various cusps. Translating well known results in the theory of adelic automorphic representations into…

We obtain density theorems for cuspidal automorphic representations of $\text{GL}_n$ over $\mathbb{Q}$ which fail the generalized Ramanujan conjecture at some place. We depart from previous approaches based on Kuznetsov-type trace formulae,…

We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of relative Langlands duality \`a la…

In this paper, we give the upper bounds on the variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series respectively. For the cusp form case, the bound comes from a large sieve inequality for symmetric cubes. We also give…

We provide an abstract multivariate central limit theorem with the Lindeberg-type error bounded in terms of Lipschitz functions (Wasserstein 1-distance) or functions with bounded second or third derivatives. The result is proved by means of…

We prove Lindel\"of-on-average upper bounds on the cubic moment of central values of $L$-functions over certain families of $\operatorname{PGL}_2/\mathbb{Q}$ automorphic representations $\pi$ given by specifying the local representation…

We prove a converse theorem for a family of L functions of degree 2 with gamma factor coming from a holomorphic cuspform. We show these L functions coincide with either those coming from a newform or a product of L functions arising from…

We obtain lower bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta function for all k > 1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the…

For $M_1$ and $ M_2$ two distinct primes, let $ H_k^\star(M_1M_2, \psi)$ denote the set of primitive newforms of level $M_1M_2$, weight $k\geq 3$ and Nebentypus $\psi$ of conductor $M_1$. Let $\pi$ be a fixed $SL(3, \mathbb{Z})$ Hecke cusp…

We obtain the $n$th centered moments of one level densities of a large orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q\sim Q$. We verify the Katz-Sarnak conjecture for these…

We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem…

In this paper, we demonstrate the existence of the second moment of the Selberg zeta function for a Fuchsian group of the first kind at $\sigma = 1$. The prime geodesic theorem plays a crucial role in this context. The proof extends to…

Let F be an orthonormal basis of weight 2 cusp forms on Gamma_0(N). We show that various weighted averages of special values L(f \tensor chi, 1) over f in F are equal to 4 pi + O(N^{-1 + epsilon}). A previous result of Duke gives an error…

There are many Rankin-Selberg integrals representing Langlands $L$-functions, and it is not apparent what the limits of the Rankin-Selberg method are. The Dimension Equation is an equality satisfied by many such integrals that suggests a…

We prove the asymptotic formulae for several moments of derivatives of GL(2) L-functions over quadratic twists. The family of L-functions we consider has root number fixed to -1 and odd orthogonal symmetry. Assuming GRH we prove the…

In a previous article we had proved an algebraicity result for the central critical value for L-functions for GL(n) x GL(n-1) over Q assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this…

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level. In contrast to previous work on…

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of…