An asymptotic formula for the logarithm of generalized partition functions

Abstract

Let $\Lambda =\{ \lambda _j\}_{j=1}^\infty$ be a set of positive integers, and let $p_\Lambda (n)$ be the number of ways of writing n as a sum of positive integers $\lambda \in \Lambda$ such that $\chi (\lambda )=1$ and $f(\lambda ) \equiv j \,\,(\mathrm {mod}\,\, m)$, where $m\ge 1$ and $j\ge 0$ are fixed integers. Here, $\chi$ and f are a certain multiplicative function and an additive function, respectively. In this paper, we obtain the asymptotic formula for $\ln \left( \sum _{n\le x} p_\Lambda (n) \right) \sim \ln p_\Lambda ([x])$ as $x\rightarrow \infty$.