Parity of the partition function

By Ken Ono. 2010

60 citations recorded by the source corpus.

Abstract Although much is known about the partition function, little is known about its parity. For the polynomials D ( x ) : = ( D x 2 + 1 ) / 24 , where D ≡ 23 ( mod 24 ) , we show that there are infinitely many m (resp. n ) for which p ( D ( m ) ) is even (resp. p ( D ( n ) ) is odd) if there is at least one such m (resp. n ). We bound the first m and n (if any) in terms of the class number h ( − D ) . For prime D we show that there are indeed infinitely many even values. To this end we construct new modular generating functions using generalized Borcherds products , and we employ Galois representations and locally nilpotent Hecke algebras.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/908df54202b645e30517053f8e2f7e1b07ed214e

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary (2001, background): “In this direction, the second author has made some progress [67], [69] towards quantifying the rarity of such congruences, and A.”