Proofs for Andrews' Conjectures 5 and 6 on $v_1(q)$

TL;DR

This paper proves Andrews' Conjectures 5 and 6 regarding the coefficients of a specific q-series, using an analysis of zeros and quadratic sequences, building on recent related proofs.

Contribution

It provides the first unconditional proof of Andrews' Conjectures 5 and 6, expanding understanding of the properties of the q-series v_1(q).

Findings

Proved Andrews' Conjectures 5 and 6.

Analyzed zeros of the trigonometric factor in the asymptotic.

Showed quadratic sequence stays away from integers infinitely often.

Abstract

Folsom, Males, Rolen, and Storzer recently proved Andrews' Conjecture~4 for the coefficients of \[ v_1(q)=\sum_{n\ge 0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}=\sum_{n\ge 0}V_1(n)q^n. \] They also proved a refined density-one version of Andrews' Conjecture~3. In this paper we prove Andrews' Conjectures~5 and~6. Our proof relies on an investigation of the simple zeros of the trigonometric factor in the Folsom--Males--Rolen--Storzer asymptotic and showing that the relevant quadratic sequence stays a positive distance from the integers infinitely often. The argument is unconditional.