# Analysis and combinatorics of partition zeta functions

## TL;DR

This paper studies partition zeta functions, providing explicit formulas, analyzing their analytic properties, and offering combinatorial proofs, thereby extending the understanding of these functions analogous to the Riemann zeta function.

## Contribution

It introduces explicit formulas and analytic insights for a family of partition zeta functions, including a combinatorial proof linking to MacMahon's partial fractions.

## Key findings

- Explicit formulas for partition zeta functions of fixed length
- Complete analysis of poles and roots of these functions
- A combinatorial proof connecting to MacMahon's decomposition

## Abstract

We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon's partial fraction decomposition of the generating function for partitions of fixed length.