Partition zeta functions

TL;DR

This paper develops new partition-theoretic formulas using transformations of q-series, infinite products, and continued fractions to compute mathematical constants and connect partition sums with zeta functions and number theory.

Contribution

It introduces novel formulas linking partitions, q-series, and zeta functions, expanding the analytical tools for number theory and mathematical constants.

Findings

Derived formulas for computing π using partition sums.

Connected partition sums to Riemann zeta and multiple zeta values.

Established new relationships between q-series transformations and number-theoretic constants.

Abstract

We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums over partitions to the Riemann zeta function, multiple zeta values, and other number-theoretic objects.