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 $H(n,k)$. 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.

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

LEONETTI, Paolo;SANNA, CARLO
2017-01-01

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 $H(n,k)$. 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.
2017
151
1
217
231
http://www.springerlink.com/content/0236-5294
https://arxiv.org/abs/1605.07424
harmonic number; p-adic valuation; Stirling number of the first kind; Mathematics (all)
Leonetti, Paolo; Sanna, Carlo
File in questo prodotto:
File Dimensione Formato  
pstirling.pdf

Accesso aperto

Descrizione: Articolo principale
Tipo di file: POSTPRINT (VERSIONE FINALE DELL’AUTORE)
Dimensione 310.79 kB
Formato Adobe PDF
310.79 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1622122
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 9
  • ???jsp.display-item.citation.isi??? 8
social impact