Rankin-Selberg coefficients in large arithmetic progressions

Archimedean Rankin-Selberg Integrals

The paper gives complete proofs of the properties of the RankinSelberg integrals for the group GL(n,R) and GL(n,C).

Basic Rankin-selberg

We present the simplest possible example of the Rankin-Selberg method, namely for a pair of holomorphic modular forms for SL(2,Z), treated independently in 1939 by Rankin and 1940 by Selberg. (Rankin has remarked that the general idea came from his advisor and mentor, Ingham.) We also recall a proof of the analytic continuation of the relevant Eisenstein series. That is, we consider the simples...

Primes in Arithmetic Progressions to Large Moduli

where <p is Euler's function and n(x) counts all primes up to x. Indeed, (1.1) is known to hold, but only under the proviso that q grow more slowly than some fixed power of logx (Siegel-Walfisz theorem). Yet, for many applications, it is important to weaken this restriction. The assumption of the Generalized Riemann Hypothesis (GRH) yields (1.1) in the much wider range q < X I / 2/ log2+e x . I...

Fourier Coefficients of Gl(n) Automorphic Forms in Arithmetic Progressions

FOURIER COEFFICIENTS OF GLpNq AUTOMORPHIC FORMS IN ARITHMETIC PROGRESSIONS

Abstract. We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GLpNq Maass cusp forms for all N > 2, satisfy a central limit theorem in a suitable range, generalizing the case N “ 2 treated by É. Fouvry, S. Ganguly, E. Kowalski and P. Michel in [4]. Such universal Gaussian behaviour relies on a deep e...