An asymptotic formula for the logarithm of generalized partition functions

  • Seiken SaitoEmail author


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 \(x\rightarrow \infty \).


Partitions Hardy–Ramanujan formula Number of prime factors 

Mathematics Subject Classification

11P82 11N37 11N60 



The author would like to thank Professor Toyokazu Hiramatsu for his encouragements and thoughtful suggestions. The author also would like to thank the referee for his or her valuable comments and careful review of this paper.


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Authors and Affiliations

  1. 1.Nagoya Bunri UniversityInazawaJapan

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