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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 2

Euler sums and non-integerness of harmonic type sums

Haydar Göral(DOKUZ EYLUL UNIVERSITY)0000-0002-8814-6295
Doğa Can Sertbaş(CUMHURIYET UNIVERSITY)0000-0002-5884-6856
Pages
—
Publication Date
2020-04-02
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Finally, we analyze generalized hyperharmonic numbers in terms of topology and relate this to non-integerness.

📌 3 Key Takeaways

🎯
Objective & Hypothesis: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.
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Key Findings & Data:Then we focus on the non-integerness of generalized hyperharmonic numbers.
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Conclusion & Impact:Finally, we analyze generalized hyperharmonic numbers in terms of topology and relate this to non-integerness.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Harmonic numbers · hyperharmonic numbers · Euler sums

Abstract

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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Cite This Article

Göral, H., Sertbaş, D. C. (2020). Euler sums and non-integerness of harmonic type sums. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 2. https://doi.org/10.15672/hujms.544489

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References (17)

  1. E. Alkan, Approximation by special values of harmonic zeta function and log-sine integrals, Commun. Number Theory Phys. (10) 7, 515–550, 2013.
  2. E. Alkan, H. Göral and D.C. Sertbaş, Hyperharmonic Numbers can Rarely be Integers, Integers, 18 Paper No. A43, 1–16, 2018.
  3. T.M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, First edition, 1976.
  4. T.M. Apostol and T.H. Vu, Dirichlet Series Related to the Riemann Zeta Function, J. Number Theory, 19, 85–102, 1984.
  5. R.C. Baker, G. Harman and J. Pintz, The difference between consecutive primes, II, Proc. Lond. Math. Soc. (3), 83, 532–562, 2001.
  6. B.C. Berndt, Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1998.
  7. D. Borwein, J.M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinb. Math. Soc. (2), 38, 277–294, 1995.
  8. K. Boyadzhiev and A. Dil, Euler sums of hyperharmonic numbers, J. Number Theory, 147, 490–498, 2015.
  9. M. Cenkçi, A. Dil and I. Mező, Evaluation of Euler-like sums via Hurwitz zeta values, Turkish J. Math. 41, 1640–1655, 2017.
  10. J.H. Conway and R.K. Guy, The Book of Numbers, Springer-Verlag, New York, 1996.
  11. H. Göral and D.C. Sertbaş, Almost all Hyperharmonic Numbers are not Integers, J. Number Theory, 171, 495–526, 2017.
  12. H. Göral and D.C. Sertbaş, A congruence for some generalized harmonic type sums, Int. J. Number Theory, 14 (4), 1033–1046, 2018.
  13. H. Göral and D.C. Sertbaş, Divisibility Properties of Hyperharmonic Numbers, Acta Math. Hungar. 154 (1), 147–186, 2018.
  14. K. Kamano, Dirichlet series associated with hyperharmonic numbers, Mem. Osaka Inst. Tech. Ser. A, 56 (2), 11–15, 2011.
  15. I. Mező, About the non-integer property of hyperharmonic numbers, Ann. Univ. Sci. Budapest. Sect. Math. 50, 13–20, 2007.
  16. W.A. Stein et. al., Sage Mathematics Software (Version 6.10.rc2), The Sage Devel- opment Team, 2015, http://www.sagemath.org.
  17. L. Theisinger, Bemerkung über die harmonische reihe, Monatsh. Math. Phys. 26, 132–134, 1915.