Distribution of parity of the partition function in arithmetic progressions

AbstractLet p(n) be the number of partitions of a non-negative integer n. In this paper we prove that for any progression r (mod t), and for large X, we have 1.(a) The number of n ≤ X such that n ≡ r (mod t) and p(n) is even is ⪢ √X, and2.(b) The number of n ≤ X such that n ≡ r (mod t) and p(n) is odd is ⪢ √Xlog X, provided that one such n exists