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New Relations Concerning a Mean Value of Some Hardy Sums and Ramanujan Sum

Year 2020, Volume: 15 Issue: 1, 148 - 157, 31.05.2020
https://doi.org/10.29233/sdufeffd.702928

Abstract

Dedekind sum first occured naturally in Dedekind’s transformation law of his eta-function. In analogy, Hardy sums are encountered in the transformation formula of the theta function. Up to now, they have many of remarkable applications in analytic number theory (Dedekind's η-function), algebraic number theory (class numbers), lattice point problems, topology and algebraic geometry. Miscellaneous arithmetical properties of these sums have been analyzed by many scholars. Recently, considering the characteristics of Hardy sums and other celebrated sums such as Ramanujan sum and Kloosterman sum, interesting and meaningful identities have been obtained. In this paper, we continue to focus on arithmetical aspects of Hardy sums and Ramanujan sum. More precisely, we consider a mean value problem of these sums and Ramanujan sum with the help of the features of Dirichlet L-functions and present some computational formulas for them.

Thanks

The author would like to thank the anonymous referees for helpful suggestions that improved the paper.

References

  • [1] T. M. Apostol, Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976, pp. 165–173.
  • [2] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory. New York: Springer-Verlag, 1976, ch. 3.
  • [3] B. C. Berndt, “Analytic Eisenstein series, theta functions and series relations in the spirit of Ramanujan,” J. Reine Angew. Math.,303/304, 332–365, 1978.
  • [4] L. Carlitz, “The reciprocity theorem of Dedekind sums,” Pac. J. Math.,3, 513–522, 1953.
  • [5] J. Chaohua, “On the mean value of Dedekind sums,” J. Number Theory,87, 173–188, 2001.
  • [6] J. B. Conrey, E. Fransen, R. Klein, and C. Scott, “Mean values of Dedekind sums,” J. Number Theory,56, 214–226, 1996.
  • [7] M. C. Dağlı, “On the hybrid mean value of generalized Dedekind sums, generalized Hardy sums and Ramanujan sum,” Accepted for publication in Bull. Math. Soc. Sci. Math. Roumanie.
  • [8] M. C. Dağlı, “On some identities involving certain Hardy sums and Kloosterman sum,” Accepted for publication in Ukr. Math. J.
  • [9] D. Han, W. Zhang, “Some new identities involving Dedekind sums and the Ramanujan sum,” Ramanujan J.,35, 253–262, 2014.
  • [10] L. A. Goldberg, “Transformations of theta-functions and analogues of Dedekind sums,” Ph.D. thesis, University of Illinois, Urbana, United States of America, 1981.
  • [11] H. Y. Liu, W. Zhang, “Some identities involving certain Hardy sums and Ramanujan sum,” Acta Math. Sin. Engl. Ser.,21, 109–116, 2005.
  • [12] W. Liu, “Mean value of Hardy sums over short intervals,” Acta Math. Acad. Paedagog. Nyházi,28, 1–11, 2012.
  • [13] W. Liu, “On the mean values of Dedekind sums over short intervals,” Notes Number Theory Discrete Math., 19 (2), 60–68, 2013.
  • [14] R. Ma, Y. L. Zhang, and M. Grützmann, “Some notes on identities for Dirichlet functions,” Acta Math. Sin. Engl. Ser.,30, 747–754, 2014.
  • [15] H. Rademacher, E. Grosswald, Dedekind sums. Washington: Math. Assoc. of America, 1972, ch. 6.
  • [16] R. Sitaramachandrarao, “Dedekind and Hardy sums,” Acta Arith.,XLIII, 325–340, 1987.
  • [17] W. Wang, D. Han, “An identity involving certain Hardy sums and Ramanujan’s sum,” Adv. Difference Equ.,261, 1– 8, 2013.
  • [18] H. Zhang, T. Zhang, “Some identities involving certain Hardy sums and general Kloosterman sums,” Mathematics, 8 (1), 1– 13, 2020.
  • [19] H. Zhang, W. Zhang, “On the identity involving certain Hardy sums and Kloosterman sums,” J. Inequal. Appl.,52, 1– 9, 2014.
  • [20] W. Zhang, “On the mean values of Dedekind sums,” J. Theor. Nombres Bordeaux,8, 429–442, 1996.

Bazı Hardy Toplamları ve Ramanujan Toplamının Ortalama Değeri Hakkında Yeni Bağıntılar

Year 2020, Volume: 15 Issue: 1, 148 - 157, 31.05.2020
https://doi.org/10.29233/sdufeffd.702928

Abstract

Dedekind toplamı, Dedekind eta-fonksiyonunun dönüşüm formülünde doğal olarak ortaya çıkmıştır. Benzer şekilde, Hardy toplamlarına ise theta fonksiyonunun dönüşüm formüllerinde karşılaşılmıştır. Günümüze kadar bu toplamlar, analitik sayılar teorisi (Dedekind η-fonksiyonu), cebirsel sayılar teorisi (sınıf sayıları), latis noktası problemleri, topoloji ve cebirsel geometri alanlarında çok sayıda önemli uygulamalara sahiptir. Bu toplamların çeşitli aritmetik özellikleri birçok bilim adamı tarafından analiz edilmiştir. Son zamanlarda ise, Hardy toplamları, Ramanujan toplamı ve Kloosterman toplamı gibi önemli toplamların sağladığı özellikler göz önüne alınarak, ilginç ve anlamlı özdeşlikler elde edilmiştir. Bu makalede, Hardy toplamları ve Ramanujan toplamına aritmetik açıdan odaklanmaya devam edeceğiz. Daha açık olarak, Dirichlet L-fonksiyonlarının özellikleri yardımıyla Hardy toplamları ve Ramanujan toplamının ortalama değer problemini ele alacağız ve onlar için bazı hesaplama formülleri sunacağız.

References

  • [1] T. M. Apostol, Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976, pp. 165–173.
  • [2] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory. New York: Springer-Verlag, 1976, ch. 3.
  • [3] B. C. Berndt, “Analytic Eisenstein series, theta functions and series relations in the spirit of Ramanujan,” J. Reine Angew. Math.,303/304, 332–365, 1978.
  • [4] L. Carlitz, “The reciprocity theorem of Dedekind sums,” Pac. J. Math.,3, 513–522, 1953.
  • [5] J. Chaohua, “On the mean value of Dedekind sums,” J. Number Theory,87, 173–188, 2001.
  • [6] J. B. Conrey, E. Fransen, R. Klein, and C. Scott, “Mean values of Dedekind sums,” J. Number Theory,56, 214–226, 1996.
  • [7] M. C. Dağlı, “On the hybrid mean value of generalized Dedekind sums, generalized Hardy sums and Ramanujan sum,” Accepted for publication in Bull. Math. Soc. Sci. Math. Roumanie.
  • [8] M. C. Dağlı, “On some identities involving certain Hardy sums and Kloosterman sum,” Accepted for publication in Ukr. Math. J.
  • [9] D. Han, W. Zhang, “Some new identities involving Dedekind sums and the Ramanujan sum,” Ramanujan J.,35, 253–262, 2014.
  • [10] L. A. Goldberg, “Transformations of theta-functions and analogues of Dedekind sums,” Ph.D. thesis, University of Illinois, Urbana, United States of America, 1981.
  • [11] H. Y. Liu, W. Zhang, “Some identities involving certain Hardy sums and Ramanujan sum,” Acta Math. Sin. Engl. Ser.,21, 109–116, 2005.
  • [12] W. Liu, “Mean value of Hardy sums over short intervals,” Acta Math. Acad. Paedagog. Nyházi,28, 1–11, 2012.
  • [13] W. Liu, “On the mean values of Dedekind sums over short intervals,” Notes Number Theory Discrete Math., 19 (2), 60–68, 2013.
  • [14] R. Ma, Y. L. Zhang, and M. Grützmann, “Some notes on identities for Dirichlet functions,” Acta Math. Sin. Engl. Ser.,30, 747–754, 2014.
  • [15] H. Rademacher, E. Grosswald, Dedekind sums. Washington: Math. Assoc. of America, 1972, ch. 6.
  • [16] R. Sitaramachandrarao, “Dedekind and Hardy sums,” Acta Arith.,XLIII, 325–340, 1987.
  • [17] W. Wang, D. Han, “An identity involving certain Hardy sums and Ramanujan’s sum,” Adv. Difference Equ.,261, 1– 8, 2013.
  • [18] H. Zhang, T. Zhang, “Some identities involving certain Hardy sums and general Kloosterman sums,” Mathematics, 8 (1), 1– 13, 2020.
  • [19] H. Zhang, W. Zhang, “On the identity involving certain Hardy sums and Kloosterman sums,” J. Inequal. Appl.,52, 1– 9, 2014.
  • [20] W. Zhang, “On the mean values of Dedekind sums,” J. Theor. Nombres Bordeaux,8, 429–442, 1996.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Makaleler
Authors

Muhammet Cihat Dağlı 0000-0003-2859-902X

Publication Date May 31, 2020
Published in Issue Year 2020 Volume: 15 Issue: 1

Cite

IEEE M. C. Dağlı, “New Relations Concerning a Mean Value of Some Hardy Sums and Ramanujan Sum”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 15, no. 1, pp. 148–157, 2020, doi: 10.29233/sdufeffd.702928.