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 GLm(QA) and GLm′(QA), respectively. 2000 Mathematics Subject Classification: 11F70, 11M26, 11M41.