On the Number of Hecke Eigenvalues of 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 Generalised Ramanujan--Petersson Conjecture, that for almost all $x\sim X$ the short interval $[x,x+H]$ contains a subset $\mathcal{S}$ (resp. ${\mathcal{S}}^{\prime })$ of size $\gg H(\log X{)}^{1/n^2-1}$ such that $A(m,1,...,1)$ is positive (resp. negative) for all $m\in \mathcal{S}$ (resp. $m\in {\mathcal{S}}^{\prime }$), provided that $(\log X{)}^{1-1/n^2}\ll H\ll X$. We also prove a slightly stronger result unconditionally for ${\mathrm{GL}}_2$ and ${\mathrm{GL}}_3$ Hecke--Maass cusp forms. In addition, we obtain results under weaker bounds towards Generalised Ramanujan--Petersson Conjecture. Finally, as a by-product of our methods, we improve earlier bounds for the number of Hecke eigenvalues of same sign also in long intervals unconditionally for ${\mathrm{GL}}_2$ and ${\mathrm{GL}}_3$ Hecke--Maass cusp forms, and under Generalised Ramanujan--Petersson Conjecture for ${\mathrm{GL}}_n$ forms when $n\ge 4$.

24 pages