<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Given a subset <italic>A</italic> of the natural numbers 𝕅 = {0, 1, 2, ···} (resp. of the ring 𝕑<italic>/ N</italic>𝕑 of residue classes modulo a positive integer <italic>N</italic>), we introduce certain sums of roots of unity associated with <italic>A</italic>. We study some of their properties, and we use them to obtain new expressions for the classical functions that characterize <italic>A</italic>, i.e. of the representation function, the counting function and the characteristic function of <italic>A</italic>. We also give an example of computations of the representation function using such expressions.</p></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.

Cyclotomic Expressions for Representation Functions

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Cyclotomic Expressions for Representation Functions"}

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...

Cyclotomic Expressions for Representation Functions

Published Online: Mar 27, 2020

Page range: 123 - 140

Received: Aug 20, 2018

Accepted: Nov 05, 2018

DOI: https://doi.org/10.2478/udt-2019-0008

Keywords
Representation function, exponential sums, cyclotomic fields

© 2019 Charles Helou, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Cyclotomic Expressions for Representation Functions

Published Online: Mar 27, 2020

Page range: 123 - 140

Received: Aug 20, 2018

Accepted: Nov 05, 2018

DOI: https://doi.org/10.2478/udt-2019-0008

KeywordsRepresentation function, exponential sums, cyclotomic fields

© 2019 Charles Helou, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Representation function, exponential sums, cyclotomic fields