Year 2020,
Volume: 49 Issue: 2, 586 - 598, 02.04.2020
Haydar Göral
,
Doğa Can Sertbaş
References
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integrals, Commun. Number Theory Phys. (10) 7, 515–550, 2013.
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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.
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opment Team, 2015, http://www.sagemath.org.
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132–134, 1915.
Euler sums and non-integerness of harmonic type sums
Year 2020,
Volume: 49 Issue: 2, 586 - 598, 02.04.2020
Haydar Göral
,
Doğa Can Sertbaş
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.