Distribution of parity of the partition function in arithmetic progressions

Let 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 (a) The nimber of n less than or equal to X such that n = r (mod t) and (b) The number of n less than or equal to X such that n =rp(n) is even is much greater than root X and r (mod t) and p(n) is odd is much greater than root X log X, provided that one such n exists.