The distribution of values of the partition function in residue classes

AbstractMaking use of an identity of Euler's involving the partition function p(n), Kolberg (Math. Scand. 7 (1959), 377–378) showed that p(n) assumes both even and odd values infinitely often. His method admits of refinement, and as a consequence we are led to the following more comprehensive result: Let q ⩾ 2, 0 ⩽ r < q and denote by Er,q(N) the number of positive integers n⩽ N such that p(n)  r (mod q). The there exist at least two distinct values of r such that, for all sufficiently large N,Er,q(N) > log log Nq log 2