On the density of the odd values of the partition function and the t-multipartition function

A folklore conjecture on the partition function asserts that the density of odd values of p(n) is 1/2. In general, for a positive integer t, let p(t)(n) be the t-multipartition function and delta(t) be the density of the odd values of p(t)(n). It is widely believed that delta(t) exists. Given an odd integer a and an integer b depending on a and t, Judge and Zanello framed an infinite family of conjectural congruence relations on p(t)(an +b) (mod 2) which establishes a striking connection between delta(a) and delta(1). As a special case t = 1, it implies that delta(1) > 0 if (3, a) = 1 and delta(a) > 0. This conjecture was proved for several values of a by Judge, Keith and Zanello. In this paper we prove that the conjecture is true for a = l(alpha) is a prime power with l >= 5 and a = 3. (C) 2021 Elsevier Inc. All rights reserved.