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

RESUMO

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 $n\geq 2$ 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. $\mathcal S')$ 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. $m\in\mathcal S'$), 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 $n\geq 4$.