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APPLICATIONS OF CLASS NUMBERS AND BERNOULLI NUMBERS TO HARMONIC TYPE SUMS
Goral, Haydar,Sertbas, Doga Can Korean Mathematical Society 2021 대한수학회보 Vol.58 No.6
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.
THE DIFFERENCE OF HYPERHARMONIC NUMBERS VIA GEOMETRIC AND ANALYTIC METHODS
Altuntas, Cagatay,Goral, Haydar,Sertbas, Doga Can Korean Mathematical Society 2022 대한수학회지 Vol.59 No.6
Our motivation in this note is to find equal hyperharmonic numbers of different orders. In particular, we deal with the integerness property of the difference of hyperharmonic numbers. Inspired by finiteness results from arithmetic geometry, we see that, under some extra assumption, there are only finitely many pairs of orders for two hyperharmonic numbers of fixed indices to have a certain rational difference. Moreover, using analytic techniques, we get that almost all differences are not integers. On the contrary, we also obtain that there are infinitely many order values where the corresponding differences are integers.