Parity results concerning the generalized divisor function involving small prime factors of integers

TL;DR

This paper investigates the asymptotic behavior of a sum involving the number of small prime factors of integers, revealing a key difference based on whether a specific number is prime or composite, using advanced analytical methods.

Contribution

It provides new asymptotic results for sums over integers with restricted prime factors, highlighting the impact of primality on the behavior of these sums.

Findings

Asymptotic formulas for $S_{-k}(x,y)$ when $k+2 \\leq y \\leq x$

Distinct asymptotic behavior depending on whether $k+1$ is prime or composite

Application of Buchstab-de Bruijn recurrence and Perron contour integral methods

Abstract

Let $\nu_y(n)$ denote the number of distinct prime factors of $n$ that are $<y$. For $k$ a positive integer, and for $k+2\leq y\leq x$, let $S_{-k}(x,y)$ denote the sum \begin{eqnarray*} S_{-k}(x,y):=\sum_{n\leq x}(-k)^{\nu_y(n)}. \end{eqnarray*} In this paper, we describe our recent results on the asymptotic behavior of $S_{-k}(x,y)$ for $k+2\leq y\leq x$, 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 M\"{o}bius function over integers with restricted prime factors and on a multiplicative generalization of the sieve.