On the fourth moment of Dirichlet L-functions: revisiting Soundararajan’s approach

Abstract

Moments of L-functions are central objects in analytic number theory, providing insight into the size, distribution, and arithmetic behavior of L-functions. This report presents a detailed and self-contained exposition of Soundararajan’s asymptotic formula for the fourth moment of Dirichlet L-functions associated with primitive Dirichlet characters modulo q at the central point s=1/2, valid uniformly for all moduli, including large highly composite ones. The proof relies on a splitting technique applied to the approximate functional equation. The diagonal terms of the short range sum yield the leading asymptotic of order q(log q)⁴, while the remaining contributions are shown to be of lower order using analytic estimates. The report develops the necessary background, establishes key lemmas, and carefully analyzes both main and error terms. This work provides a clear and accessible treatment of a significant result in the theory of L-functions, serving both as a reference for Soundararajan’s method and as an introduction to the analytic techniques underlying higher moments of Dirichlet L-functions.