On the parity of the partition function

Let 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 vertical bar D(x)vertical bar 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 >= n(0), then vertical bar D(x)vertical bar >= log x/log2-logn(0)/log2. Moreover, if D is an infinite subset of N such that p(D, n) is odd for all n >= n(0) and n(0) >= min{d: d is an element of D}, then vertical bar D(x)vertical bar >= log x / log2 - logn(0)/log2. These lower bounds are essentially the best possible. (c) 2006 Elsevier Inc. All rights reserved.