Small Values of Coefficients of a Half Lerch Sum

TL;DR

This paper investigates the coefficients of a specific q-hypergeometric series related to partitions and quadratic fields, proposing conjectures and partial results on their distribution, inspired by prior work on Ramanujan's series.

Contribution

It introduces new conjectures on the coefficients of a Lerch sum-based q-series and provides partial results towards understanding their distribution, extending previous work on Ramanujan's series.

Findings

Partial results on the distribution of small coefficient values.

Formulation of new conjectures inspired by Andrews's work.

Connection of coefficients to quadratic field arithmetic.

Abstract

Andrews, Dyson and Hickerson proved many interesting properties of coefficients for a Ramanujan's $q$-hypergeometric series by relating it to real quadratic field $\Q(\sqrt{6})$ and using the arithmetic of $\Q(\sqrt{6})$, hence solved a conjecture of Andrews on the distributions of its Fourier coefficients. Motivated by Andrews's conjecture, we discuss an interesting $q$-hypergeometric series which comes from a Lerch sum and rank and crank moments for partitions and overpartitions. We give Andrews-like conjectures for its coefficients. We obtain partial results on the distributions of small values of its coefficients toward these conjectures.