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ON ELLIPTIC ANALOGUE OF THE HARDY SUMS
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 Title & Authors
ON ELLIPTIC ANALOGUE OF THE HARDY SUMS
Simsek, Yilmaz; Kim, Dae-Yeoul; Koo, Ja-Kyung;
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 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
Dedekind sums;Hardy sums;Eisenstein series;theta functions;Weierstrass -function;Jacobi form;
 Language
English
 Cited by
1.
Special functions related to Dedekind-type DC-sums and their applications, Russian Journal of Mathematical Physics, 2010, 17, 4, 495  crossref(new windwow)
 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. crossref(new window)

3.
A. Bayad, Sommes elliptiques multiples d'Apostol-Dedekind-Zagier, C. R. Math. Acad. Sci. Paris 339 (2004), no. 7, 457-462. crossref(new window)

4.
B. C. Berndt, Generalized Dedekind eta-functions and generalized Dedekind sums, Trans. Amer. Math. Soc. 178 (1973), 495-508. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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.

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. crossref(new window)

9.
D. Bertrand, Theta functions and transcendence, Ramanujan J. 1 (1997), no. 4, 339-350. crossref(new window)

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. crossref(new window)

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. crossref(new window)

17.
J. Lewittes, Analytic continuation of Eisenstein series, Trans. Amer. Math. Soc. 171 (1972), 469-490. crossref(new window)

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. crossref(new window)

20.
H. Rademacher, Zur Theorie der Modulfunktionen, J. Reine Angew. Math. 167 (1932), 312-336. crossref(new window)

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.

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. crossref(new window)

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. crossref(new window)