Applications of Class Numbers and Bernoulli Numbers to Harmonic Type Sums
Bull. Korean Math. Soc. Published online July 7, 2021
Haydar Goral and Doga Can Sertbas
Izmir Institute of Technology; Cukurova 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