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

Abstract We prove asymptotic formulas for the complex coefficients of $$(\zeta q;q)_\infty ^{-1}$$ ( ζ q ; ) ∞ - 1 , where $$\zeta $$ is a root unity, and apply our results to determine secondary terms in asymptotics p ( b n ), number integer partitions with parts congruent modulo . Our imply that, as $$n \rightarrow \infty n → difference $$p(a_1,b,n)-p(a_2,b,n)$$ p a , b 2 $$a_1 \ne a_2$$ ≠ oscillates like cosine when renormalized by elementary functions. Moreover, we give arbitrary linear combinations $$\{p(a,b,n)\}_{1 \le b}.$$ { } ≤ .