Almost all hyperharmonic numbers are not integers

Göral, Haydar; Sertbaş, Doğa Can

doi:10.1016/j.jnt.2016.07.023

Cited by 13 publications

(17 citation statements)

References 7 publications

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“…Later this result was improved by Amrane and Belbachir [1,2] (see also Cereceda [8]) to some general class of the parameter r . All these results were further sharpened by Göral and Sertbaş [14].…”

Section: On the Noninteger Propertymentioning

confidence: 74%

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Evaluation of Euler-like sums via Hurwitz zeta values

Dil¹,

Mező²,

Cenkci³

2017

Turk J Math

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In this paper we collect two generalizations of harmonic numbers (namely generalized harmonic numbers and hyperharmonic numbers) under one roof. Recursion relations, closed-form evaluations, and generating functions of this unified extension are obtained. In light of this notion we evaluate some particular values of Euler sums in terms of odd zeta values. We also consider the noninteger property and some arithmetical aspects of this unified extension.

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Section: On the Noninteger Propertymentioning

confidence: 74%

“…This conjecture was justified for a class of pairs (n, r) by AitAmrane and Belbachir [1,2] and Cereceda [8]. Very recently Göral and Sertbaş [14] extended the current results for large orders; considering primes in short intervals, they proved that almost all hyperharmonic numbers are not integers.…”

Section: Hyperharmonic Numbers: Starting With Hmentioning

confidence: 83%

Evaluation of Euler-like sums via Hurwitz zeta values

Dil¹,

Mező²,

Cenkci³

2017

Turk J Math

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“…The case r = 1 was already proved by Theisinger [17]. Based on three different approaches, namely analytic, combinatorial and algebraic, the authors [11] proved that almost all hyperharmonic numbers are not integers. This yields an almost answer to Mező's problem [15].…”

Section: Introductionmentioning

confidence: 92%

“…This yields an almost answer to Mező's problem [15]. Moreover, in the same paper [11], it was deduced that if n is even or a prime power, or r is odd then the corresponding hyperharmonic number is not integer.…”

Section: Introductionmentioning

confidence: 94%

Euler sums and non-integerness of harmonic type sums

Göral

Sertbaş

2020

Hacettepe Journal of Mathematics and Statistics

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We show that Euler sums of generalized hyperharmonic numbers can be evaluated in terms of Euler sums of generalized harmonic numbers and special values of the Riemann zeta function. Then we focus on the non-integerness of generalized hyperharmonic numbers. We prove that almost all generalized hyperharmonic numbers are not integers and our error term is sharp and the best possible. Finally, we analyze generalized hyperharmonic numbers in terms of topology and relate this to non-integerness.

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“…For details and historical introductions, please see [1, 3-6, 8, 11, 16, 19-23] and references therein. From [7,9,10,12,15], we know that the classical hyperharmonic numbers are defined by…”

Section: Introductionmentioning

confidence: 99%

Computation and theory of Euler sums of generalized hyperharmonic numbers

2018

Comptes Rendus. Mathématique

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Recently, Dil and Boyadzhiev [10] proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence ({0} r , 1). In this paper we show that the sums of multiple harmonic numbers whose indices are the sequence {0} r , 1; {1} k−1 can be expressed in terms of (multiple) zeta values, multiple harmonic numbers and Stirling numbers of the first kind, and give an explicit formula.

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Almost all hyperharmonic numbers are not integers

Cited by 13 publications

References 7 publications

Evaluation of Euler-like sums via Hurwitz zeta values

Evaluation of Euler-like sums via Hurwitz zeta values

Euler sums and non-integerness of harmonic type sums

Computation and theory of Euler sums of generalized hyperharmonic numbers

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