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 $$x\rightarrow \infty $$ .