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Year 2020, Volume: 49 Issue: 2, 586 - 598, 02.04.2020
https://doi.org/10.15672/hujms.544489

Abstract

References

  • [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.

Euler sums and non-integerness of harmonic type sums

Year 2020, Volume: 49 Issue: 2, 586 - 598, 02.04.2020
https://doi.org/10.15672/hujms.544489

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.

References

  • [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.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Haydar Göral 0000-0002-8814-6295

Doğa Can Sertbaş This is me 0000-0002-5884-6856

Publication Date April 2, 2020
Published in Issue Year 2020 Volume: 49 Issue: 2

Cite

APA Göral, H., & Sertbaş, D. C. (2020). Euler sums and non-integerness of harmonic type sums. Hacettepe Journal of Mathematics and Statistics, 49(2), 586-598. https://doi.org/10.15672/hujms.544489
AMA Göral H, Sertbaş DC. Euler sums and non-integerness of harmonic type sums. Hacettepe Journal of Mathematics and Statistics. April 2020;49(2):586-598. doi:10.15672/hujms.544489
Chicago Göral, Haydar, and Doğa Can Sertbaş. “Euler Sums and Non-Integerness of Harmonic Type Sums”. Hacettepe Journal of Mathematics and Statistics 49, no. 2 (April 2020): 586-98. https://doi.org/10.15672/hujms.544489.
EndNote Göral H, Sertbaş DC (April 1, 2020) Euler sums and non-integerness of harmonic type sums. Hacettepe Journal of Mathematics and Statistics 49 2 586–598.
IEEE H. Göral and D. C. Sertbaş, “Euler sums and non-integerness of harmonic type sums”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, pp. 586–598, 2020, doi: 10.15672/hujms.544489.
ISNAD Göral, Haydar - Sertbaş, Doğa Can. “Euler Sums and Non-Integerness of Harmonic Type Sums”. Hacettepe Journal of Mathematics and Statistics 49/2 (April 2020), 586-598. https://doi.org/10.15672/hujms.544489.
JAMA Göral H, Sertbaş DC. Euler sums and non-integerness of harmonic type sums. Hacettepe Journal of Mathematics and Statistics. 2020;49:586–598.
MLA Göral, Haydar and Doğa Can Sertbaş. “Euler Sums and Non-Integerness of Harmonic Type Sums”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, 2020, pp. 586-98, doi:10.15672/hujms.544489.
Vancouver Göral H, Sertbaş DC. Euler sums and non-integerness of harmonic type sums. Hacettepe Journal of Mathematics and Statistics. 2020;49(2):586-98.