ON THE p-ADIC VALUATION OF GENERALIZED HARMONIC NUMBERS

Altuntaş Ç.

Bulletin of the Korean Mathematical Society, vol.60, no.4, pp.933-955, 2023 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 60 Issue: 4
  • Publication Date: 2023
  • Doi Number: 10.4134/bkms.b220399
  • Journal Name: Bulletin of the Korean Mathematical Society
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.933-955
  • Keywords: generalized harmonic numbers, Harmonic numbers, p-adic valuation
  • Istanbul Technical University Affiliated: Yes

Abstract

For any prime number p, let J(p) be the set of positive integers n such that the numerator of the nth harmonic number in the lowest terms is divisible by this prime number p. We consider an extension of this set to the generalized harmonic numbers, which are a natural extension of the harmonic numbers. Then, we present an upper bound for the number of elements in this set. Moreover, we state an explicit condition to show the finiteness of our set, together with relations to Bernoulli and Euler numbers.