Standard zero-free regions for Rankin\u2013Selberg L-functions via sieve theory

Peter Humphries,Farrell Brumley

doi:10.1007/s00209-018-2136-8

Peter Humphries, Farrell Brumley

Open Access

https://doi.org/10.1007/s00209-018-2136-8

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Abstract

We give a simple proof of a standard zero-free region in the t-aspect for the Rankin–Selberg L-function L(s,pi times widetilde{pi }) for any unitary cuspidal automorphic representation pi of mathrm {GL}_n(mathbb {A}_F) that is tempered at every nonarchimedean place outside a set of Dirichlet density zero.

Highlights

Let F be a number field, let n be a positive integer, and let π be a unitary cuspidal automorphic representation of GLn(AF ) with L-function L(s, π ), with π normalised such that its central character is trivial on the diagonally embedded copy of the positive reals
For s = σ + it, and this generalises to a zero-free region for L(s, π) of the form σ
Remark 1.3 Note that in [12, Exercise 4, p. 108], it is claimed that one can use this method to prove a zero-free region similar to (1.1) when π π and π π ; the hint to this exercise is invalid, as the Dirichlet coefficients of the logarithmic derivative of the auxiliary L-function suggested in this hint are real but not necessarily nonpositive. (In particular, as stated, [12, Exercise 4, p. 108] would imply the nonexistence of Landau–Siegel zeroes upon taking f to be a quadratic Dirichlet character and g to be the trivial character.)

Summary

Introduction

Let F be a number field, let n be a positive integer, and let π be a unitary cuspidal automorphic representation of GLn(AF ) with L-function L(s, π ), with π normalised such that its central character is trivial on the diagonally embedded copy of the positive reals. The proof of the prime number theorem due to de la Valleé–Poussin gives a zero-free region for the Riemann zeta function ζ (s) of the form σ 108], it is claimed that one can use this method to prove a zero-free region similar to (1.1) when π π and π π ; the hint to this exercise is invalid, as the Dirichlet coefficients of the logarithmic derivative of the auxiliary L-function suggested in this hint are real but not necessarily nonpositive.

Results

Conclusion