Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions

Abdelmejid Bayad[1]; Yilmaz Simsek[2]

  • [1] Université d’Evry Val d’Essonne Département de mathématiques Bd. F. Mitterrand, 91025 Evry Cedex (France)
  • [2] University of Akdeniz Faculty of Arts and Science Department of Mathematics 07058 Antalya (Turkey)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 5, page 1977-1993
  • ISSN: 0373-0956

Abstract

top
In this paper we study three new shifted sums of Apostol-Dedekind-Rademacher type. The first sums are written in terms of Jacobi modular forms, and the second sums in terms of cotangent and the third sums are expressed in terms of special values of the Barnes multiple zeta functions. These sums generalize the classical Dedekind-Rademacher sums. The main aim of this paper is to state and prove the Dedekind reciprocity laws satisfied by these new sums. As an application of our Dedekind reciprocity law we show how to derive all the well-known results on Dedekind reciprocity law studied by Hall-Wilson-Zagier, Beck-Berndt-Dieter, Katayama and Nagasaka-Ota-Sekine.

How to cite

top

Bayad, Abdelmejid, and Simsek, Yilmaz. "Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions." Annales de l’institut Fourier 61.5 (2011): 1977-1993. <http://eudml.org/doc/219708>.

@article{Bayad2011,
abstract = {In this paper we study three new shifted sums of Apostol-Dedekind-Rademacher type. The first sums are written in terms of Jacobi modular forms, and the second sums in terms of cotangent and the third sums are expressed in terms of special values of the Barnes multiple zeta functions. These sums generalize the classical Dedekind-Rademacher sums. The main aim of this paper is to state and prove the Dedekind reciprocity laws satisfied by these new sums. As an application of our Dedekind reciprocity law we show how to derive all the well-known results on Dedekind reciprocity law studied by Hall-Wilson-Zagier, Beck-Berndt-Dieter, Katayama and Nagasaka-Ota-Sekine.},
affiliation = {Université d’Evry Val d’Essonne Département de mathématiques Bd. F. Mitterrand, 91025 Evry Cedex (France); University of Akdeniz Faculty of Arts and Science Department of Mathematics 07058 Antalya (Turkey)},
author = {Bayad, Abdelmejid, Simsek, Yilmaz},
journal = {Annales de l’institut Fourier},
keywords = {Elliptic Dedekind sums; modular forms; theta functions; ellpitic functions; Bernoulli functions; Jacobi modular forms; elliptic Dedekind sums},
language = {eng},
number = {5},
pages = {1977-1993},
publisher = {Association des Annales de l’institut Fourier},
title = {Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions},
url = {http://eudml.org/doc/219708},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Bayad, Abdelmejid
AU - Simsek, Yilmaz
TI - Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 5
SP - 1977
EP - 1993
AB - In this paper we study three new shifted sums of Apostol-Dedekind-Rademacher type. The first sums are written in terms of Jacobi modular forms, and the second sums in terms of cotangent and the third sums are expressed in terms of special values of the Barnes multiple zeta functions. These sums generalize the classical Dedekind-Rademacher sums. The main aim of this paper is to state and prove the Dedekind reciprocity laws satisfied by these new sums. As an application of our Dedekind reciprocity law we show how to derive all the well-known results on Dedekind reciprocity law studied by Hall-Wilson-Zagier, Beck-Berndt-Dieter, Katayama and Nagasaka-Ota-Sekine.
LA - eng
KW - Elliptic Dedekind sums; modular forms; theta functions; ellpitic functions; Bernoulli functions; Jacobi modular forms; elliptic Dedekind sums
UR - http://eudml.org/doc/219708
ER -

References

top
  1. A. Bayad, Jacobi forms in two variables: Multiple elliptic Dedekind sums, The Kummer-von Staudt Clausen Congruences for elliptic Bernoulli functions and values of Hecke L-functions 
  2. A. Bayad, Sommes de Dedekind elliptiques et formes de Jacobi, Ann. Instit. Fourier 51 Fasc. 1, (2001), 29-42 Zbl1034.11030MR1821066
  3. A. Bayad, Applications aux sommes elliptiques multiples d’Apostol-Dedekind-Zagier, C.R.A.S Paris, Ser. I 339 fascicule 8, Série I, (2004), 539-532 Zbl1099.11025MR2099541
  4. A. Bayad, Sommes elliptiques multiples d’Apostol-Dedekind-Zagier, C.R.A.S Paris, Ser. I 339 fascicule 7, Série I, (2004), 457-462 Zbl1099.11026MR2099541
  5. M. Beck, Dedekind cotangent sums, Acta Arithmetica 109 (2003), 109-130 Zbl1061.11043MR1980640
  6. B. C. Berndt, Reciprocity theorems for Dedekind sums and generalizations, Adv. in Math. 23 (1977), 285-316 Zbl0342.10014MR429711
  7. B. C. Berndt, U. Dieter, Sums involving the greatest integer function and Riemann-Stieltjes integration, J. reine angew. Math. 337 (1982), 208-220 Zbl0487.10002MR676053
  8. U. Dieter, Cotangent sums, a further generalization of Dedekind sums, J. Number Th. 18 (1984), 289-305 Zbl0537.10005MR746865
  9. R. R. Hall, J. C. Wilson, D. Zagier, Reciprocity formulae for general Dedekind-Rademacher sums, Acta Arith. 73 (1995), 389-396 Zbl0847.11020MR1366045
  10. K. Katayama, Barne’s Double zeta function, the Dedekind Sum and Ramanujan’s Formula, Tokyo J. Math. 27 (2004), 41-56 Zbl1075.11028MR2060073
  11. K. Katayama, Barne’s Multiple function and Apostol’s Generalized Dedekind Sum, Tokyo J. Math. 27 (2004), 57-74 Zbl1075.11029MR2060074
  12. Y. Nagasaka, K. Ota, C. Sekine, Generalizations of Dedekind sums and their reciprocity laws, acta Arith 106 (2003), 355-378 Zbl1055.11031MR1957911
  13. K. Ota, Derivatives of Dedekind sums and their reciprocity law, Journal of Number Theory 98 (2003), 280-309 Zbl1038.11028MR1955418
  14. S. N. M. Ruijsenaars, On Barnes’ Multiple Zeta and Gamma Functions, Advances in Mathematics 156 (2000), 107-132 Zbl0966.33013MR1800255
  15. R. Sczech, Dedekindsummen mit elliptischen Funktionen, Invent.math 76 (1984), 523-551 Zbl0521.10021MR746541
  16. A. Weil, Elliptic functions according to Eisenstein and Kronecker, (1976), Springer-Verlag Zbl0318.33004MR562289

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.