论文标题
某些ETA晶状体系数的均衡
Parity of the coefficients of certain eta-quotients
论文作者
论文摘要
我们研究了某些ETA品质的系数的均等，并广泛研究了$ m $ groumard分区的情况。我们的定理涉及其奇数的密度，尤其是为指定系数建立空白模量2；自相似度模拟2;以及算术进展中的一致性家族。对于所有$ m \ leq 28 $，我们要么建立这些类型的新结果，因此不知道以前的结果，要么猜想这种结果是不可能的。 我们所有的工作都与我们为任意ETA质品提出的新的总体猜想一致，这极大地扩展了Parkin-Shanks对分区功能的经典猜想。我们在本文中提出了其他几个开放性问题，并通过建议在该领域进行未来研究的特定研究方向清单来结束。
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.
