<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

Publié en ligne: 27 mars 2020

Pages: 123 - 140

Reçu: 20 août 2018

Accepté: 05 nov. 2018

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

Mots clés
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

Publié en ligne: 27 mars 2020

Pages: 123 - 140

Reçu: 20 août 2018

Accepté: 05 nov. 2018

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

Mots clésRepresentation 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.

Mots clés
Representation function, exponential sums, cyclotomic fields