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.
Categories: math.NT