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Arithmetic Partitions of Natural Numbers, Zeta Regularization, Prime Divisibility and Dirichlet L-Functions
Subject area: Science,Engineering and Technology · Area of research: Number Theory
DOI: 10.64388/ire1724687
Abstract
The partition of the positive integers into arithmetic progressions provides a useful framework for studying divergent series through analytic continuation. This paper investigates arithmetic partitions of the natural numbers and their relationship with the Riemann zeta function, Hurwitz zeta function, Bernoulli polynomials, prime divisibility and Dirichlet L-functions. The ordinary series of positive integers diverges, but its zeta-regularized value is ζ(−1) = −1/12. We derive formulas for regularized sums of residue classes modulo an integer, examine partitions determined by prime divisibility, and obtain weighted identities involving Dirichlet characters. We extend the method to higher powers and discuss an infinite partition based on powers of two. Particular attention is given to the distinction between finite additivity and countable additivity in regularized summation.
Keywords
Arithmetic partitions; Riemann zeta function; Hurwitz zeta function; zeta regularization; Bernoulli polynomials; prime numbers; Dirichlet characters; Dirichlet L-functions
References
[1] Apostol TM. Introduction to analytic number theory. New York: Springer; 1976. https://doi.org/10.1007/978-1-4757-5579-4
[2] Edwards HM. Riemann's zeta function. New York: Academic Press; 1974.
[3] Olver FWJ, Olde Daalhuis AB, Lozier DW, Schneider BI, Boisvert RF, Clark CW, et al., editors. NIST Digital Library of Mathematical Functions, Chapter 25: Zeta and related functions [Internet]. Gaithersburg (MD): National Institute of Standards and Technology. Available from: https://dlmf.nist.gov/25
[4] Washington LC. Introduction to cyclotomic fields. 2nd ed. New York: Springer; 1997. https://doi.org/10.1007/978-1-4612-1934-7
[5] Whittaker ET, Watson GN. A course of modern analysis. 4th ed. Cambridge: Cambridge University Press; 1927.
[6] Apostol TM. Modular functions and Dirichlet series in number theory. 2nd ed. New York: Springer; 1990. https://doi.org/10.1007/978-1-4612-0999-7
[7] Berndt BC. The determination of the values of the Dirichlet L-functions. J Number Theory. 1972;4:375–395.
How to cite this paper
@article{1724687,
author = {Dr. K. V. Rama Rao},
title = {Arithmetic Partitions of Natural Numbers, Zeta Regularization, Prime Divisibility and Dirichlet L-Functions},
journal = {Iconic Research And Engineering Journals},
year = {2026},
volume = {10},
number = {5},
pages = {952-958},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1724687.pdf},
abstract = {The partition of the positive integers into arithmetic progressions provides a useful framework for studying divergent series through analytic continuation. This paper investigates arithmetic partitions of the natural numbers and their relationship with the Riemann zeta function, Hurwitz zeta function, Bernoulli polynomials, prime divisibility and Dirichlet L-functions. The ordinary series of positive integers diverges, but its zeta-regularized value is ζ(−1) = −1/12. We derive formulas for regularized sums of residue classes modulo an integer, examine partitions determined by prime divisibility, and obtain weighted identities involving Dirichlet characters. We extend the method to higher powers and discuss an infinite partition based on powers of two. Particular attention is given to the distinction between finite additivity and countable additivity in regularized summation.},
keywords = {Arithmetic partitions; Riemann zeta function; Hurwitz zeta function; zeta regularization; Bernoulli polynomials; prime numbers; Dirichlet characters; Dirichlet L-functions},
month = {November},
doi = {https://doi.org/10.64388/ire1724687}
}