Parity of the coefficients of certain eta-quotients

Document Type

Article

Publication Date

1-1-2021

Department

Department of Mathematical Sciences

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≤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.

Publication Title

Journal of Number Theory

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