Bull. Korean Math. Soc. 2021; 58(6): 1463-1481
Online first article July 7, 2021 Printed November 30, 2021
https://doi.org/10.4134/BKMS.b201045
Copyright © The Korean Mathematical Society.
Haydar G\"{o}ral, Do\u{g}a Can Sertba\c{s}
Izmir Institute of Technology; \c{C}ukurova University
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
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