Computing Dirichlet character sums to a power-full modulus

TL;DR

This paper presents an efficient method for computing Dirichlet character sums and L-functions for smooth moduli, leveraging the Postnikov formula and quadratic exponential sums to achieve power savings and improved complexity.

Contribution

The paper introduces a novel approach combining the Postnikov formula with an analytic algorithm to compute Dirichlet character sums and L-functions more efficiently for smooth moduli.

Findings

Achieves power savings for smooth moduli

Develops a method to compute L-functions with complexity exponent 1/3

Provides a practical algorithm for large-scale computations

Abstract

The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an analytic algorithm previously derived by the author. This leads to a power-saving if the modulus is smooth enough. As an application, a fast, and potentially practical, method to compute Dirichlet L-functions with complexity exponent 1/3 for smooth enough moduli is derived.