Asymptotic formulae for mixed congruence stacks

Abstract

Much like the important work of Hardy and Ramanujan (Proc Lond Math Soc 2(17):75–115, 1919) proving the asymptotic formula for the partition function, Auluck (Math Proc Camb Philos Soc 47:679–686, 1951) and Wright (Quart J Math (Oxf) 22:107–116, 1971) gave similar formulas for unimodal sequences. Following the circle method of Wright, we provide the asymptotic expansion for unimodal sequences on a two-parameter family of mixed congruence relations, with parts on one side up to the peak satisfying $$r \pmod {m}$$ and parts on the other side $$-r\pmod {m}$$ . Techniques used in the proofs include Wright’s circle method, modular transformations, and bounding of complex integrals.