On the Number of Hecke Eigenvalues of the Same Sign on ${\mathrm{GL}}_n$

Abstract

We study the distribution of signs of the (real-valued) Hecke eigenvalues $A(m,1,...,1)$ of self-dual Hecke--Maass cusp forms for the group ${\mathrm{SL}}_n(\mathbb{Z})$, where $n\ge 2$ is an integer. Our main result establishes, under the Generalised Ramanujan--Petersson Conjecture, that for all $X\le x\le 2X$ outside a set of measure $o(X)$ the short interval $(x,x+H]$ contains at least $H(\log X{)}^{1/n^2-1}$ indices with both positive and negative Hecke eigenvalues, provided that $(\log X{)}^{1-1/n^2+\epsilon }\le H\le X/10$. For ${\mathrm{GL}}_2$ Hecke--Maass cusp forms we obtain a stronger bound unconditionally, and for self-dual ${\mathrm{GL}}_3$ forms we obtain the same logarithmic exponent unconditionally. We also establish results under weaker bounds towards the Generalised Ramanujan--Petersson Conjecture together with the Generalised Lindelöf Hypothesis, and obtain lower bounds for the number of sign changes and for the number of Hecke eigenvalues of each sign in long intervals.

arXiv commentSome updates and corrections; 26 pages