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arXiv · 2003.06621

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

Moni Kumari·Jyoti Sengupta
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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, $λ_f(n)$ and $λ_g(n)$, and we may normalize $f$ and $g$ so that $λ_f(1)=1=λ_g(1)$. In this article, we first prove that the sequence $\{λ_f(n)λ_g(n)\}_{n \in \mathbb{N}}$ 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 $g$.

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Moni Kumari, Jyoti Sengupta. 2020-03-14. The first simultaneous sign change for Fourier coefficients of Hecke-Maass forms. https://doi.org/10.1007/s11139-020-00268-9

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