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ON THE p-ADIC VALUATION OF GENERALIZED HARMONIC NUMBERS

Cagatay Altuntas
대한수학회 (The Korean Mathematical Society)|2023.07|대한수학회보 (Bulletin of the KMS)|제60권 제4호|pp. 933-955|DOI 10.4134/BKMS.b220399

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참고문헌

키워드

Harmonic numbersp-adic valuationgeneralized harmonic numbers
주제분류수학

초록

For any prime number $p$, let $J(p)$ be the set of positive integers $n$ such that the numerator of the $n^{th}$ 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.

Abstract

For any prime number $p$, let $J(p)$ be the set of positive integers $n$ such that the numerator of the $n^{th}$ 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.

참고문헌 (23)

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