On the parity of the partition function

Abstract

AbstractLet N be the set of all positive integers and D a subset of N. Let p(D,n) be the number of partitions of n with parts in D and let |D(x)| denote the number of elements of D not exceeding x. It is proved that if D is an infinite subset of N such that p(D,n) is even for all n⩾n0, then |D(x)|⩾logx/log2−logn0/log2. Moreover, if D is an infinite subset of N such that p(D,n) is odd for all n⩾n0 and n0⩾min{d:d∈D}, then |D(x)|⩾logx/log2−logn0/log2. These lower bounds are essentially the best possible