Asymptotics of sums of divisor functions over sequences with restricted factorization structure

Rafael Jakimczuk and Matilde Lalín
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 617–634
DOI: 10.7546/nntdm.2022.28.4.617-634
Full paper (PDF, 235 Kb)

Details

Authors and affiliations

Rafael Jakimczuk
División Matemática, Universidad Nacional de Luján
Buenos Aires, Argentina

Matilde Lalín
Département de mathématiques et de statistique, Université de Montréal
CP 6128, succ. Centre-ville, Montreal, QC H3C 3J7, Canada

Abstract

We compute asymptotics of the sums of general divisor functions over h-free numbers, h-full numbers and other arithmetically interesting sets and conditions. The main tool for obtaining these results is Perron’s formula.

Keywords

• Divisor function
• Sum of divisors function
• h-free number
• h-full number
• Perfect powers
• h-free part
• h-full part

2020 Mathematics Subject Classification

• 11A25
• 11B99
• 11N37

References

1. Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers, sixth ed. Oxford University Press, Oxford. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles.
2. Huxley, M. N. (2003). Exponential sums and lattice points. III. Proceedings of the London Mathematical Society, 87(3), 591–609.
3. Ivić, A. (1985). The Riemann Zeta-function. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.
4. Ram Murty, M. (2008). Problems in Analytic Number Theory, second ed. Graduate Texts in Mathematics, Vol. 206, Springer, New York.
5. Walfisz, A. (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin.

Manuscript history

• Received: 23 May 2022
• Revised: 29 September 2022
• Accepted: 12 October 2022
• Online First: 14 October 2022