Computational Number Theory in Relation with L-Functions

TL;DR

This paper explores theoretical and practical methods for computing L-functions, including local and global cases, with a focus on numerical techniques and special sums like Gauss and Jacobi sums.

Contribution

It introduces new computational methods and detailed analyses for L-functions, enhancing both theoretical understanding and practical computation.

Findings

Development of methods for counting points over finite fields

Analysis of inverse Mellin transforms for Dirichlet L-functions

Introduction of useful numerical techniques for L-function computation

Abstract

We give a number of theoretical and practical methods related to the computation of L-functions, both in the local case (counting points on varieties over finite fields, involving in particular a detailed study of Gauss and Jacobi sums), and in the global case (for instance Dirichlet L-functions, involving in particular the study of inverse Mellin transforms); we also give a number of little-known but very useful numerical methods, usually but not always related to the computation of L-functions.