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 | 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 , t...

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Bibliographic Details
Published in:Journal of number theory 2007-02, Vol.122 (2), p.283-289
Main Authors: Dai, Li-Xia, Chen, Yong-Gao
Format: Article
Language:English
Subjects:
Lower bound
Parity
Partition function
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Summary: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 2 − log 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 2 − log n 0 / log 2 . These lower bounds are essentially the best possible.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2006.05.015