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

Abstract

For all integers $n\ge k\ge 1$, define $H(n,k):=\sum 1/(i_1\cdots i_k)$, where the sum is extended over all positive integers $i_1<\cdots <i_k\le n$. 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 Erdős-Niven and Chen-Tang, we study the $p$-adic valuation of $H(n,k)$. In particular, for any prime number $p$, integer $k\ge 2$, and $x\ge (k-1)p$, 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.

12 pages, 3 figures