Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON ELLIPTIC ANALOGUE OF THE HARDY SUMS

  • Simsek, Yilmaz (DEPARTMENT OF MATHEMATICS FACULTY OF ARTS AND SCIENCE UNIVERSITY OF AKDENIZ) ;
  • Kim, Dae-Yeoul (NATIONAL INSTITUTE FOR MATHEMATICAL SCIENCE) ;
  • Koo, Ja-Kyung (DEPARTMENT OF MATHEMATICAL SCIENCES KOREA ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY)
  • Published : 2009.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Main purpose of this paper is to define an elliptic analogue of the Hardy sums. Some results, which are related to elliptic analogue of the Hardy sums, are given.

Keywords

References

  1. T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer- Verlag, 1976.
  2. A. Bayad, Applications aux sommes elliptiques d'Apostol-Dedekind-Zagier, C. R. Math. Acad. Sci. Paris 339 (2004), no. 8, 529-532. https://doi.org/10.1016/j.crma.2004.07.019
  3. A. Bayad, Sommes elliptiques multiples d'Apostol-Dedekind-Zagier, C. R. Math. Acad. Sci. Paris 339 (2004), no. 7, 457-462. https://doi.org/10.1016/j.crma.2004.07.018
  4. B. C. Berndt, Generalized Dedekind eta-functions and generalized Dedekind sums, Trans. Amer. Math. Soc. 178 (1973), 495-508. https://doi.org/10.2307/1996714
  5. B. C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan, J. Reine Angew. Math. 303/304 (1978), 332-365. https://doi.org/10.1515/crll.1978.303-304.332
  6. B. C. Berndt and L. A. Goldberg, Analytic properties of arithmetic sums arising in the theory of the classical theta functions, SIAM J. Math. Anal. 15 (1984), no. 1, 143-150. https://doi.org/10.1137/0515011
  7. B. C. Berndt and L. Schoenfeld, Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applications to number theory, Acta Arith. 28 (1975/76), no. 1, 23-68. https://doi.org/10.4064/aa-28-1-23-68
  8. B. C. Berndt and B. P. Yeap, Explicit evaluations and reciprocity theorems for finite trigonometric sums, Adv. in Appl. Math. 29 (2002), no. 3, 358-385. https://doi.org/10.1016/S0196-8858(02)00020-9
  9. D. Bertrand, Theta functions and transcendence, Ramanujan J. 1 (1997), no. 4, 339-350. https://doi.org/10.1023/A:1009749608672
  10. M. Can, M. Cenkci, and V. Kurt, Generalized Hardy-Berndt sums, Proc. Jangjeon Math. Soc. 9 (2006), no. 1, 19-38.
  11. U. Dieter, Cotangent sums, a further generalization of Dedekind sums, J. Number Theory 18 (1984), no. 3, 289-305. https://doi.org/10.1016/0022-314X(84)90063-5
  12. S. Egami, An elliptic analogue of the multiple Dedekind sums, Compositio Math. 99 (1995), no. 1, 99-103.
  13. M. I. Knoop, Modular Functions in Analytic Number Theory, Markham Publishing Company, Chicago, 1970.
  14. N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York, 1993.
  15. V. Kurt, On Dedekind sums, Indian J. Pure Appl. Math. 21 (1990), no. 10, 893-896.
  16. J. Lewittes, Analytic continuation of the series $\sum(m+nz)^{-s}$, Trans. Amer. Math. Soc.159 (1971), 505-509. https://doi.org/10.2307/1996026
  17. J. Lewittes, Analytic continuation of Eisenstein series, Trans. Amer. Math. Soc. 171 (1972), 469-490. https://doi.org/10.2307/1996391
  18. S. J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Cambridge University Press, Cambridge, 1988.
  19. M. R. Pettet and R. Sitaramachandrarao, Three-term relations for Hardy sums, J. Number Theory 25 (1987), no. 3, 328-339. https://doi.org/10.1016/0022-314X(87)90036-9
  20. H. Rademacher, Zur Theorie der Modulfunktionen, J. Reine Angew. Math. 167 (1932), 312-336. https://doi.org/10.1515/crll.1932.167.312
  21. H. Rademacher, Topics in Analytic Number Theory, Springer-Verlag, 1988.
  22. R. A. Rankin, Elementary proofs of relations between Eisenstein series, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 2, 107-117. https://doi.org/10.1017/S0308210500013925
  23. B. Schoneneberg, Zur Theorie der verallgemeinerten Dedekindschen Modulfunktionen, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II (1969), 119-128.
  24. Y. Simsek, Theorems on three-term relations for Hardy sums, Turkish J. Math. 22 (1998), no. 2, 153-162.
  25. Y. Simsek, Relations between theta-functions Hardy sums Eisenstein and Lambert series in the transformation formula of $log_{g,h}$(z), J. Number Theory 99 (2003), no. 2, 338-360. https://doi.org/10.1016/S0022-314X(02)00072-0
  26. Y. Simsek, On Weierstrass $\wp$(z) function Hardy sums and Eisenstein series, Proc. Jangjeon Math. Soc. 7 (2004), no. 2, 99-108.
  27. Y. Simsek, Generalized Dedekind sums associated with the Abel sum and the Eisenstein and Lambert series, Adv. Stud. Contemp. Math. (Kyungshang) 9 (2004), no. 2, 125-137.
  28. Y. Simsek and S. Yang, Transformation of four Titchmarsh-type infinite integrals and generalized Dedekind sums associated with Lambert series, Adv. Stud. Contemp. Math. (Kyungshang) 9 (2004), no. 2, 195-202.
  29. R. Sitaramachandrarao, Dedekind and Hardy sums, Acta Arith. XLVIII (1978), 325-340.
  30. M. Waldschmidt, P. Moussa, J. M. Luck, and C. Itzykson, From Number Theory to Physics, Springer-Verlag, 1995.
  31. D. Zagier, Higher dimensional Dedekind sums, Math. Ann. 202 (1973), 149-172. https://doi.org/10.1007/BF01351173

Cited by

  1. Special functions related to Dedekind-type DC-sums and their applications vol.17, pp.4, 2010, https://doi.org/10.1134/S1061920810040114