Deducing the positive odd density of p(n) from that of a multipartition function: An unconditional proof

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Department of Mathematical Sciences


A famous conjecture of Parkin-Shanks predicts that p(n) is odd with density 1/2. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the ‘‘striking” fact that, if for any A≡±1(mod6) the multipartition function pA(n) has positive odd density, then so does p(n). Similarly, the positive odd density of any pA(n) with A≡3(mod6) would imply that of p3(n). Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture.

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Journal of Number Theory