Asymptotics for the twisted eta-product and applications to sign changes in partitions

Abstract

We prove asymptotic formulas for the complex coefficients of (ζq; q) −1 ∞ , where ζ is a root of unity, and apply our results to determine secondary terms in the asymptotics for p(a, b, n), the number of integer partitions of n with largest part congruent a modulo b. Our results imply that, as n → ∞, the difference p(a1, b, n) − p(a2, b, n) for a1 = a2 oscillates like a cosine when renormalized by elementary functions. Moreover, we give asymptotic formulas for arbitrary linear combinations of {p(a, b, n)} 1≤a≤b .2020 Mathematics Subject Classification. 11P82, 11P83. Key words and phrases. circle method, partitions, asymptotics, sign-changes, secondary terms. 1 A similar sounding, but entirely different problem is to study the total number of parts that are all congruent to a modulo b. This problem was studied in detail by [4]) for ordinary partitions and recently by Craig [10] for distinct parts partitions.