Jesse Jääsaari

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

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 i...