Zero free region for L-functions of Hecke-Maass forms

Abstract

This thesis investigates explicit zero-free regions for the L-functions of even Hecke-Maass forms. We start with Stechkin’s refined approach to the study of the classical zero-free region for the Riemann zeta function ζ(s) and McCurley’s extensions to Dirichlet L-functions. Building on these foundations, Creech, Hamieh, Khunger, Sinha, Streipel and Tsang adapted these methods to L-functions for modular Hecke newforms of even weight k with respect to Γ_0(N). We extend this framework to even Hecke-Maass forms of weight 0 with respect to SL_2(Z). Our approach combines Stechkin’s differencing technique, logarithmic derivative estimates, nonnegative trigonometric polynomials, auxiliary parameter methods, and precise bounds on gamma factors associated with appropriate L-functions. We establish explicit zero-free regions with numerical constants for L(s, Sym^m f) with m ∈ {1, 2, 3, 4}, enhancing our understanding of the location of the zeros of these L-functions.