Analysis and combinatorics of partition zeta functions

Author:

Schneider Robert1,Sills Andrew V.2ORCID

Affiliation:

1. Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA

2. Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia 30458, USA

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.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

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1. Investigation of norms of overpartitions;The Ramanujan Journal;2022-09-29

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