The first simultaneous sign change for Fourier coefficients of Hecke-Maass forms

Abstract

Let $f$ and $g$ be two Hecke-Maass cusp forms of weight zero for $S{L}_2(\mathbb{Z})$ with Laplacian eigenvalues $\frac{1}{4}+u^2$ and $\frac{1}{4}+v^2$, respectively. Then both have real Fourier coefficients say, ${\lambda }_f(n)$ and ${\lambda }_g(n)$, and we may normalize $f$ and $g$ so that ${\lambda }_f(1)=1={\lambda }_g(1)$. In this article, we first prove that the sequence $\{{\lambda }_f(n){\lambda }_g(n){\}}_{n\in \mathbb{N}}$ has infinitely many sign changes. Then we derive a bound for the first negative coefficient for the same sequence in terms of the Laplacian eigenvalues of $f$ and $g$.

To appear in the Ramanujan Journal