On the density of the odd values of the partition function, II: An infinite conjectural framework

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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.

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© 2018 Elsevier Inc. All rights reserved. Publisher’s version of record: https://doi.org/10.1016/j.jnt.2018.01.016

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Journal of Number Theory