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arXiv · 2010.09881

Parity of the coefficients of certain eta-quotients

William J. Keith·Fabrizio Zanello
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Abstract

We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of $m$-regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all $m \leq 28$, we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible. All of our work is consistent with a new, overarching conjecture that we present for arbitrary eta-quotients, greatly extending Parkin-Shanks' classical conjecture for the partition function. We pose several other open questions throughout the paper, and conclude by suggesting a list of specific research directions for future investigations in this area.

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William J. Keith, Fabrizio Zanello. 2021-07-13. Parity of the coefficients of certain eta-quotients. https://arxiv.org/abs/2010.09881

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