Overview
This study focuses on an omega result concerning the initial sign change of coefficients within symmetric power $L$-functions associated with Hecke-Maass cusp forms. Specifically, it extends prior work by Lamzouri regarding the lower bound for the first positive integer $n_f$ where the Hecke eigenvalue $\lambda_f(n_f)$ deviates from zero. The current research applies this extension to Hecke-Maass forms, without reliance on the Generalized Ramanujan Conjecture, and establishes analogous results for the coefficients of their symmetric power $L$-functions.
Research Context
Lamzouri previously demonstrated a lower bound for the least positive integer $n_f$ such that the Hecke eigenvalue $\lambda_f(n_f) \neq 0$. This earlier finding applied to many holomorphic Hecke cusp forms of even integral weight $k$, indicating that $n_f \ge (\log k)^{1-o(1)}$. The current research builds upon this foundational result.
Approach
The research extends Lamzouri's established result from holomorphic Hecke cusp forms to the domain of Hecke-Maass forms. A notable aspect of this extension is that it proceeds without requiring the assumption of the Generalized Ramanujan Conjecture. Furthermore, the methodology is applied to investigate the coefficients of symmetric power $L$-functions of Hecke-Maass forms, aiming to determine if analogous results hold in this context.
Findings
- The study extends Lamzouri's lower bound result for the least positive integer $n_f$ (where $\lambda_f(n_f) \neq 0$) to Hecke-Maass forms.
- This extension for Hecke-Maass forms is achieved without assuming the Generalized Ramanujan Conjecture.
- Similar results were proven to hold for the coefficients of symmetric power $L$-functions when associated with Hecke-Maass forms.