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

[摘要] 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 equivalent to +/- 1 ( mod 6) 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 equivalent to 3 ( mod 6) 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. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Partition function;Density odd values;Multipartition function;Partition identity;Binary q-series