Cyclotomic Expressions for Representation Functions

Charles Helou

doi:10.2478/udt-2019-0008

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Cyclotomic Expressions for Representation Functions

Uniform distribution theory

Volume 14 (2019): Issue 1 (June 2019)

By: Charles Helou

Open Access

|Mar 2020

Abstract

Given a subset A of the natural numbers 𝕅 = {0, 1, 2, ···} (resp. of the ring 𝕑/ N𝕑 of residue classes modulo a positive integer N), we introduce certain sums of roots of unity associated with A. We study some of their properties, and we use them to obtain new expressions for the classical functions that characterize A, i.e. of the representation function, the counting function and the characteristic function of A. We also give an example of computations of the representation function using such expressions.

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Articles in this issue

DOI: https://doi.org/10.2478/udt-2019-0008 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

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Language: English

Page range: 123 - 140

Submitted on: Aug 20, 2018

Accepted on: Nov 5, 2018

Published on: Mar 27, 2020

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

Representation function,

exponential sums,

cyclotomic fields

Related subjects:

Mathematics,

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2020 Charles Helou, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 14 (2019): Issue 1 (June 2019)