On the p-adic valuation of Stirling numbers of the first kind

Abstract

For all integers $n \geq k \geq 1$, define $H(n,k) := \sum 1 / (i_1 \cdots i_k)$, where the sum is extended over all positive integers $i_1 < \cdots < i_k \leq n$. These quantities are closely related to the Stirling numbers of the first kind by the identity $H(n,k) = s(n + 1, k + 1) / n!$. Motivated by the works of Erd\H{o}s-Niven and Chen-Tang, we study the $p$-adic valuation of $H(n,k)$. In particular, for any prime number $p$, integer $k \geq 2$, and $x \geq (k-1)p$, we prove that $\nu_p(H(n,k)) < -(k - 1)(\log_p(n/(k - 1)) - 1)$ for all positive integers $n \in [(k-1)p, x]$ whose base $p$ representations start with the base $p$ representation of $k - 1$, but at most $3x^{0.835}$ exceptions. We also generalize a result of Lengyel by giving a description of $\nu_2(H(n,2))$ in terms of an infinite binary sequence.