Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2021; 58(6): 1463-1481

Published online November 30, 2021 https://doi.org/10.4134/BKMS.b201045

Copyright © The Korean Mathematical Society.

Applications of class numbers and Bernoulli numbers to harmonic type sums

Haydar G\"{o}ral, Do\u{g}a Can Sertba\c{s}

Izmir Institute of Technology; \c{C}ukurova University

Abstract

Divisibility properties of harmonic numbers by a prime number $p$ have been a recurrent topic. However, finding the exact $p$-adic orders of them is not easy. Using class numbers of number fields and Bernoulli numbers, we compute the exact $p$-adic orders of harmonic type sums. Moreover, we obtain an asymptotic formula for generalized harmonic numbers whose $p$-adic orders are exactly one.

Keywords: Harmonic numbers, Bernoulli numbers, regular primes, class number

MSC numbers: Primary 11B83, 11B68, 5A10