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 $m\le 28$, 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.

我们研究了某些 eta 商的系数的奇偶性，广泛研究了m正则分区的情况。我们的定理涉及奇数值的密度，特别是为指定系数建立模 2 的空隙；自相似性模 2；和无限的等差级数同余族。对所有人$米\le 28$，我们要么在未知的情况下建立这些类型的新结果，要么扩展以前的结果，或者推测这样的结果是不可能的。

我们所有的工作都与我们针对任意 eta 商提出的一个新的总体猜想相一致，极大地扩展了 Parkin-Shanks 对配分函数的经典猜想。我们在整篇论文中提出了其他几个未解决的问题，最后提出了该领域未来调查的具体研究方向清单。