Perron’s Formula and the Prime Number Theorem for Automorphic L-Functions

Abstract

In this paper the classical Perron's formula is modified so that it now depends no longer on sizes of individual terms but on a sum over a short interval. When applied to automorphic L-functions, this new Perron's formula may allow one to avoid estimation of individual Fourier coefficients, without assuming the Generalized Ramanujan Conjecture (GRC). As an application, a prime number theorem for Rankin-Selberg L-functions L(s, π ×π ) is proved unconditionally without assuming GRC, where π and π are automorphic irreducible cuspidal representations of GL m (Q A ) and GL m (Q A ), respectively.

On the Selberg orthogonality for automorphic L-functions

“…If π π, the Rankin-Selberg L-function need not have non-negative coefficients and Problem 1 is more difficult to solve. Under the assumption that at least one of representations π and π is self-contragradient, Liu et al [9] have proved that…”

Resonance between automorphic forms and exponential functions

“…In other words, we seek upper bounds and main terms for 3) between two automorphic cuspidal representations π and π appears as the prime number theorem for the Rankin-Selberg L-function L(s, π ×π ) as proved by Liu-Wang-Ye [8], Liu-Ye [10], and Wu-Ye [15]. Recall that…”

Moments of the products of quadratic twists of automorphic L-functions

“…Moreover, if m = 2 and π gives a normalized holomorphic Hecke eigenform of level 1, by a consequence of the prime number theorem for automorphic forms (see [10]) and the relation a π ( p 2 ) = a π ( p) 2 − 2, Hypothesis F holds. Now, the main theorem is stated as follows: Then, for any k 1 , .…”