Parity Results Concerning the Generalized Divisor Function Involving Small Prime Factors of Integers

Authors

• Krishnaswami Alladi
• Ankush Goswami, The University of Texas Rio Grande Valley

Document Type

Conference Proceeding

Publication Date

7-2-2026

Abstract

Let νy(n) denote the number of distinct prime factors of n that are k a positive integer, and for k+2≤y≤x, let S−k(x,y) denote the sum S−k(x,y):=∑n≤x(−k)νy(n).

In this paper, we describe our recent results on the asymptotic behavior of S−k(x,y) for k+2≤y≤x, and x sufficiently large. There is a crucial difference in the asymptotic behavior of S−k(x,y) when k+1 is a prime and k+1 is composite, and this makes the problem particularly interesting. The results are derived utilizing a combination of the Buchstab-de Bruijn recurrence, the Perron contour integral method, and certain difference-differential equations. We present a summary of our results against the background of earlier work of the first author on sums of the Möbius function over integers with restricted prime factors and on a multiplicative generalization of the sieve.

Comments

https://rdcu.be/fx2kc

Recommended Citation

Alladi, Krishnaswami, and Ankush Goswami. “Parity Results Concerning the Generalized Divisor Function Involving Small Prime Factors of Integers.” In Some Contributions to Number Theory and Beyond, edited by Baskar Balasubramanyam, Kaneenika Sinha, and Mathukumalli Vidyasagar. Springer Nature Switzerland, 2026. https://doi.org/10.1007/978-3-032-10837-1_8.

Publication Title

Some Contributions to Number Theory and Beyond

DOI

10.1007/978-3-032-10837-1_8