Jayashree Kalita

Affiliation: Vanderbilt University

Title Of Talk: From a Conjecture of Andrews to Almost Alternating Sign Patterns

Abstract: Computer experiments led Andrews, in 1986, to conjecture striking sign patterns and growth phenomena for the coefficients of five partition-theoretic q-series from the Ramanujan's Lost Notebook. The first of these functions, the now-famous series \[ \sigma(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q;q)_n} \] exhibits remarkable growth and vanishing behavior, which was proven by Andrews, Dyson, and Hickerson, by tying this series to the arithmetic of the quadratic field $\mathbb{Q}(\sqrt{6})$. Cohen further uncovered that the numerical phenomenon was due to the q-series being what we would now call, thanks to work of Lewis-Zagier, a period integral of a Maass waveform. This example also foreshadowed the modern theories of mock Maass theta functions initiated by Zwegers, and quantum modular forms introduced by Zagier. \\ However, the other four q-series remained largely unexplored until recent work of Folsom, Males, Rolen, and Storzer, who proved some of the Andrews’ conjectures for the series \[ v_1(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}. \] Jointly with Kundu, Storzer and Wang, we established almost alternating sign patterns for coefficients of the remaining three q-series along with proving a conjecture of Andrews from his 1986 paper. Using analytic techniques such as the method of steepest descent and the circle method, we derived asymptotics for the coefficients, whose alternating and oscillatory behavior explains the observed patterns. We also introduced a new family of q-series exhibiting similar phenomena. In this talk, I will give a non-technical overview of the main ideas.

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