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Some Results on Harmonic Type Sums

Year 2020, Volume: 8 Issue: 1, 642 - 653, 31.01.2020

Haydar Göral , Doğa Can Sertbaş

https://doi.org/10.29130/dubited.622285
Cited By: 1

Abstract

In this study, we consider the summatory function of convolutions of the
Möbius function with harmonic numbers, and we show that these summatory
functions are linked to the distribution of prime numbers. In particular, we
give infinitely many asymptotics which are consequences of the Riemann
hypothesis. We also give quantitative estimate for the moment function which
counts non-integer hyperharmonic numbers. Then, we obtain the asymptotic
behaviour of hyperharmonics.

Keywords

Möbius function hyperharmonic numbers Dirichlet series

Thanks

We thank to the anonymous referee for the suggestions which improved the quality of the paper.

References

  • [1] E. Alkan, H. Göral, D. C. Sertbaş, “Hyperharmonic numbers can be rarely integers”, Integers, vol. 18, no. A43, 2018.
  • [2] T. M. Apostol, Introduction to Analytic Number Theory, 1st ed., New York, US: Springer-Verlag, 1976.
  • [3] J. H. Conway, R. K. Guy, The Book of Numbers, New York, US: Springer-Verlag, 1996.
  • [4] H. Davenport, Graduate Texts in Mathematics: Mutliplicative Number Theory, 3rd ed., New York, US: Springer, 2000.
  • [5] H. Göral, D. C. Sertbaş “Almost all Hyperharmonic Numbers are not Integers”, Journal of Number Theory, vol. 171, pp. 495-526, 2017.
  • [6] S. Ikehara, “An extension of Landau’s theorem in the analytic theory of numbers”, J. Math. and Phys. M.I.T., vol. 10, pp. 1-12, 1931.
  • [7] L. Theisinger, “Bemerkung über die harmonische reihe”, Monatshefte für Mathematik und Physik, vol. 26, pp. 132–134, 1915.

Harmonik Tipi Toplamlar Üzerine Bazı Sonuçlar

Year 2020, Volume: 8 Issue: 1, 642 - 653, 31.01.2020

Haydar Göral , Doğa Can Sertbaş

https://doi.org/10.29130/dubited.622285
Cited By: 1

Abstract

Bu çalışmada, Möbius
fonksiyonunun harmonik sayılarla konvolüsyonunun toplamsal fonksiyonunu ele
alacağız ve bu toplamsal fonksiyonun asalların dağılımı ile ilişkili olduğunu
göstereceğiz. Özel olarak, Riemann hipotezinin sonucu olan sonsuz çoklukta
asimptotik vereceğiz. Ayrıca tamsayı olmayan hiperharmoniklerin sayaç fonksiyonunun
momentleri için niceliksel bir kestirim vereceğiz. Sonra da hiperharmoniklerin
asimptotik davranışını elde edeceğiz.

Keywords

Möbius fonksiyonu hiperharmonik sayılar Dirichlet serileri

References

  • [1] E. Alkan, H. Göral, D. C. Sertbaş, “Hyperharmonic numbers can be rarely integers”, Integers, vol. 18, no. A43, 2018.
  • [2] T. M. Apostol, Introduction to Analytic Number Theory, 1st ed., New York, US: Springer-Verlag, 1976.
  • [3] J. H. Conway, R. K. Guy, The Book of Numbers, New York, US: Springer-Verlag, 1996.
  • [4] H. Davenport, Graduate Texts in Mathematics: Mutliplicative Number Theory, 3rd ed., New York, US: Springer, 2000.
  • [5] H. Göral, D. C. Sertbaş “Almost all Hyperharmonic Numbers are not Integers”, Journal of Number Theory, vol. 171, pp. 495-526, 2017.
  • [6] S. Ikehara, “An extension of Landau’s theorem in the analytic theory of numbers”, J. Math. and Phys. M.I.T., vol. 10, pp. 1-12, 1931.
  • [7] L. Theisinger, “Bemerkung über die harmonische reihe”, Monatshefte für Mathematik und Physik, vol. 26, pp. 132–134, 1915.
There are 7 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Haydar Göral 0000-0002-8814-6295

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

Publication Date January 31, 2020
Published in Issue Year 2020 Volume: 8 Issue: 1

Cite

APA Göral, H., & Sertbaş, D. C. (2020). Some Results on Harmonic Type Sums. Duzce University Journal of Science and Technology, 8(1), 642-653. https://doi.org/10.29130/dubited.622285
AMA Göral H, Sertbaş DC. Some Results on Harmonic Type Sums. DUBİTED. January 2020;8(1):642-653. doi:10.29130/dubited.622285
Chicago Göral, Haydar, and Doğa Can Sertbaş. “Some Results on Harmonic Type Sums”. Duzce University Journal of Science and Technology 8, no. 1 (January 2020): 642-53. https://doi.org/10.29130/dubited.622285.
EndNote Göral H, Sertbaş DC (January 1, 2020) Some Results on Harmonic Type Sums. Duzce University Journal of Science and Technology 8 1 642–653.
IEEE H. Göral and D. C. Sertbaş, “Some Results on Harmonic Type Sums”, DUBİTED, vol. 8, no. 1, pp. 642–653, 2020, doi: 10.29130/dubited.622285.
ISNAD Göral, Haydar - Sertbaş, Doğa Can. “Some Results on Harmonic Type Sums”. Duzce University Journal of Science and Technology 8/1 (January 2020), 642-653. https://doi.org/10.29130/dubited.622285.
JAMA Göral H, Sertbaş DC. Some Results on Harmonic Type Sums. DUBİTED. 2020;8:642–653.
MLA Göral, Haydar and Doğa Can Sertbaş. “Some Results on Harmonic Type Sums”. Duzce University Journal of Science and Technology, vol. 8, no. 1, 2020, pp. 642-53, doi:10.29130/dubited.622285.
Vancouver Göral H, Sertbaş DC. Some Results on Harmonic Type Sums. DUBİTED. 2020;8(1):642-53.

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