On a Multiplicative Partition Function

Abstract

Let $D(s)=\sum^\infty_{m=1}a_mm^{-s}$ be the Dirichlet series generated by the infinite product $\prod^\infty_{k=2}(1-k^{-s})$. The value of $a_m$ for squarefree integers $m$ with $n$ prime factors depends only on the number $n$, and we let $f(n)$ denote this value. We prove an asymptotic estimate for $f(n)$ which allows us to solve several problems raised in a recent paper by M. V. Subbarao and A. Verma.