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 $SL_2(\mathbb Z)$ 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 $g$.