On the density of the odd values of the partition function

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The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely, the conjecture that the partition function p(n) is equidistributed modulo 2.

Our main result will relate the densities, say, δt, of the odd values of the t-multipartition functions pt(n), for several integers t. In particular, we will show that if δt > 0 for some t ∈ {5, 7, 11, 13, 17, 19, 23, 25}, then (assuming it exists) δ1> 0; that is, p(n) itself is odd with positive density. Notice that, currently, the best unconditional result does not even imply that p(n) is odd for √x values of n ≤ x. In general, we conjecture that δt= 1/2 for all t odd, i.e., that similarly to the case of p(n), all multipartition functions are in fact equidistributed modulo 2.

Our arguments will employ a number of algebraic and analytic methods, ranging from an investigation modulo 2 of some classical Ramanujan identities and several other eta product results, to a unified approach that studies the parity of the Fourier coefficients of a broad class of modular form identities recently introduced by Radu.

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© Springer Nature Switzerland AG 2018. Publisher’s version of record: https://doi.org/10.1007/s00026-018-0397-x

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Annals of Combinatorics