On the density of the odd values of the partition function, II: An infinite conjectural framework
Document Type
Article
Publication Date
7-2018
Abstract
We continue our study of a basic but seemingly intractable problem in integer partition theory, namely the conjecture that p(n) is odd exactly 50% of the time. Here, we greatly extend on our previous paper by providing a doubly-indexed, infinite framework of conjectural identities modulo 2, and show how to, in principle, prove each such identity. However, our conjecture remains open in full generality.
A striking consequence is that, under suitable existence conditions, if any t-multipartition function is odd with positive density and t ≠ 0 (mod 3), then p(n) is also odd with positive density. These are all facts that appear virtually impossible to show unconditionally today.
Our arguments employ a combination of algebraic and analytic methods, including certain technical tools recently developed by Radu in his study of the parity of the Fourier coefficients of modular forms.
Publication Title
Journal of Number Theory
Recommended Citation
Judge, S.,
&
Zanello, F.
(2018).
On the density of the odd values of the partition function, II: An infinite conjectural framework.
Journal of Number Theory,
188, 357-370.
http://doi.org/10.1016/j.jnt.2018.01.016
Retrieved from: https://digitalcommons.mtu.edu/math-fp/65
Publisher's Statement
© 2018 Elsevier Inc. All rights reserved. Publisher’s version of record: https://doi.org/10.1016/j.jnt.2018.01.016