On the parity of the partition function

✍ Scribed by Li-Xia Dai; Yong-Gao Chen

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 105 KB
• Volume: 122
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.05.015

✦ Synopsis

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 |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 n 0 , then |D(x)| log x/ log 2log n 0 / log 2. 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 ∈ D}, then |D(x)| log x/ log 2log n 0 / log 2. These lower bounds are essentially the best possible.