On the parity of the partition function Original Research Article

Let image be the set of all positive integers and D a subset of image. 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 image such that p(D,n) is even for all ngreater-or-equal, slantedn0, then D(x)greater-or-equal, slantedlogx/log2−logn0/log2. Moreover, if D is an infinite subset of image such that p(D,n) is odd for all ngreater-or-equal, slantedn0 and image, then D(x)greater-or-equal, slantedlogx/log2−logn0/log2. These lower bounds are essentially the best possible.