On $v_2(q),v_3(q),v_4(q)$, and Andrews'Conjectures 5 and 6

Abstract

In this paper, we prove an exceptional sign pair phenomenon for three $q$-series from Ramanujan's Lost Notebook, namely $v_2(q),v_3(q)$, and $v_4(q)$. This was first observed by Andrews in a 1986 paper. Building on the recent work of Kundu, Storzer, Wang, and the author, which established that the coefficients of these $q$-series are alternating in sign except in a density-zero set, we study the exceptional indices where the alternating sign pattern fails. We prove that these exceptional indices occur infinitely often in a structured manner. More precisely, we show that there exist infinitely many pairs of consecutive coefficients having the same sign, and that at least one coefficient in each pair is a local minimum of the sequence of absolute values of the coefficients. Our proof combines precise asymptotic expansions for the coefficients with a careful analysis of an oscillatory factor appearing in the expansions which governs the exceptional sign behavior.