某些 eta-商系数的奇偶性，II：偶数正则分拆的情形
Parity of the coefficients of certain eta-quotients, II: The case of even-regular partitions
AI总结：
本文研究偶数 m-正则分拆函数 b_m 的系数奇偶密度，提出完整分类猜想，并利用 Ramanujan–Kolberg 恒等式得到子数列同余及 b_8 奇偶性的简化证明。
摘要：
我们继续研究 eta-商奇数值的密度，这里聚焦于 m 为偶数时的 m-正则分拆函数 b_m。基于大量计算证据，我们提出一个简洁的猜想，它尤其完整分类了这些密度：令 m=2^j m_0，其中 m_0 为奇数。若 2^j<m_0，则 b_m 的奇数密度为 1/2；此外，在每个（非常数）子等差数列 An+B 上，该密度也等于 1/2。若 2^j>m_0，则已知密度为零的 b_m 在无穷多个互不嵌套的子等差数列上恒为偶数。这一猜想以及本文的所有其他猜想，都与我们在前一工作中给出的关于 eta-商的“主猜想”相一致。总体而言，我们关于偶数 m 的 b_m 的结果所确定的行为，与 m 为奇数的情形有显著不同。同样有趣的是，在子等差数列 An+B 上，b_m 经常会在某些 t 值下与多分拆函数 p_t 的奇偶性一致。我们恰当地利用 Ramanujan–Kolberg 恒等式推出了一大类此类结果；例如，b_{28}(49n+12) ≡ p_3(7n+2) (mod 2)。进一步的推论包括针对各种 b_m 的若干“几乎总是同余式”，以及专门关于 b_{11} 的新奇偶性结果。最后，我们以一个简单得多的证明结束本文，该证明证明了 Cherubini–Mercuri 最近一篇论文的主要结果，即完整刻画 b_8 的奇偶性。
英文摘要：
We continue our study of the density of the odd values of eta-quotients, here focusing on the $m$-regular partition functions $b_m$ for $m$ even. Based on extensive computational evidence, we propose an elegant conjecture which, in particular, completely classifies such densities: Let $m = 2^j m_0$ with $m_0$ odd. If $2^j < m_0$, then the odd density of $b_m$ is $1/2$; moreover, such density is equal to $1/2$ on every (nonconstant) subprogression $An+B$. If $2^j > m_0$, then $b_m$, which is already known to have density zero, is identically even on infinitely many non-nested subprogressions. This and all other conjectures of this paper are consistent with our ''master conjecture'' on eta-quotients presented in the previous work. In general, our results on $b_m$ for $m$ even determine behaviors considerably different from the case of $m$ odd. Also interesting, it frequently happens that on subprogressions $An+B$, $b_m$ matches the parity of the multipartition functions $p_t$, for certain values of $t$. We make a suitable use of Ramanujan-Kolberg identities to deduce a large class of such results; as an example, $b_{28}(49n+12) \equiv p_3(7n+2) \pmod{2}$. Additional consequences are several ''almost always congruences'' for various $b_m$, as well as new parity results specifically for $b_{11}$. We wrap up our work with a much simpler proof of the main result of a recent paper by Cherubini-Mercuri, which fully characterized the parity of $b_8$.
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