Odd values of partition functions related to binary quadratic forms

Abstract

Let [Formula: see text] be the number of representations of [Formula: see text] by all reduced quadratic forms of discriminant [Formula: see text]. We give an asymptotic formula for the number of [Formula: see text] such that [Formula: see text] is odd, where [Formula: see text] are integers and [Formula: see text] if [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text]. As applications, we improve early results on odd values of the number of four-core partitions, the Rogers–Ramanujan’s partitions and five-regular partitions.