The sign changes of Fourier coefficients of Eisenstein series

TL;DR

This paper investigates how the signs of Fourier coefficients of Eisenstein series influence the determination of the series itself, providing both individual and statistical results that extend known theorems for cusp forms.

Contribution

It establishes new theorems on the extent to which sign patterns of Eisenstein series' Fourier coefficients determine the series, including a strong multiplicity-one type result.

Findings

Sign patterns of Fourier coefficients can determine Eisenstein series.

Frequency of initial sign sequences of Fourier coefficients is characterized.

Eisenstein series are uniquely identified by sign data on primes with density > 1/2.

Abstract

In this paper we prove a number of theorems that determine the extent to which the signs of the Hecke eigenvalues of an Eisenstein newform determine the newform. We address this problem broadly and provide theorems of both individual and statistical nature. Many of these results are Eisenstein series analogues of well-known theorems for cusp forms. For instance, we determine how often the p-th Fourier coefficients of an Eisenstein newform begin with a fixed sequence of signs \varepsilon_p = {\pm 1, 0}. Moreover, we prove the following variant of the strong multiplicity-one theorem: an Eisenstein newform is uniquely determined by the signs of its Hecke eigenvalues with respect to any set of primes with density greater than 1/2.