On the Hecke Eigenvalues of Maass Forms

TL;DR

This paper proves the existence of primes with specific properties related to Hecke eigenvalues of Maass forms and estimates their density, advancing understanding of their distribution and spectral behavior.

Contribution

It establishes bounds on primes with certain eigenvalue properties and provides a lower bound on their natural density, a novel result in the spectral theory of Maass forms.

Findings

Existence of primes p with |_p|=|_p|=1 and p  (N(1+|t_|))^c.

Explicit constant c=0.27332 for the prime bound.

Lower density bound of 34/35 for such primes.

Abstract

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the Satake parameters of $\phi$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $0.27332$. In addition, we prove that the natural density of such primes $p$ ($p\nmid N$ and $|\alpha_{p}|=|\beta_{p}| = 1$) is at least $34/35$.