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

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• 作者：P. Leonetti ; C. Sanna
• 关键词：Key words and phrasesStirling number of the first kind ; harmonic number ; p ; adic valuation
• 刊名：Acta Mathematica Hungarica
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：151
• 期：1
• 页码：217-231
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Netherlands
• ISSN：1588-2632
• 卷排序：151

文摘

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 Erdős–Niven and Chen–Tang, we study the p-adic valuation of H(n, k). Lengyel proved that \({\nu_p(H(n, k)) > -k\log_p n+O_k(1)}\) and we conjecture that there exists a positive constant c = c(p, k) such that \({\nu_p(H(n, k)) < -c\log n}\) for all large n. In this respect, we prove the conjecture in the affirmative for all \({n \leq x}\) whose base p representations start with the base p representation of k − 1, but at most \({3 x^{0.835}}\) 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.