On efficient computation of sums of characters on the basis of A. G. Postnikov methods

Nikolaj Glazunov

doi:10.15407/fmmit2021.32.019

Abstract

An efficient p-adic method and the structure of an algorithm for computing the sums of characters of finite abelian groups are presented. The method and algorithm are based on the A.G. Postnikov summation method of characters modulo a prime power and its developments. A brief survey of the theory of characters of finite abelian groups, p-adic arithmetic and analysis is presented. Questions of the efficiency of p-adic methods are discussed. Moreover, we present results of computation of other types of sums of characters (Kloosterman sums), which are connecting with Artin-Schreier coverings over prime finite fields. The corresponding method and algorithm are based on the development of another method by A.G. Postnikov. Examples of computation of sums of characters are given.

Highlights

The characters of abelian groups underlie Pontryagin's duality theory [1]
Postnikov's method of summetion of characters modulo a prime power we present p-adic method and algorithm of computing the sums of characters of finite abelian groups
We present results on computation of Kloosterman sums by extension of other A.G

Summary

Introduction

The characters of abelian groups underlie Pontryagin's duality theory [1]. Such characters find applications in mathematics, in the theory of computation, in physics, and in their applications. The fields and rings of p - adic numbers have found important and interesting applications in number theory, algebra, mathematical physics, in the computations, and in other branches of science and applications (see [2,3,4,5,6,7,8] and references therein). On the basis of these considerations and A.G. Postnikov's method of summetion of characters modulo a prime power we present p-adic method and algorithm of computing the sums of characters of finite abelian groups. We present results on computation of Kloosterman sums by extension of other A.G. Postnikov method, which is presented in [7].

Characters of abelian groups

On p - adic analisys

Sums of Dirichlet characters and the method of Postnikov indg m

Algorithm and computations