On multiple analogues of Ramanujan’s formulas for certain Dirichlet series

Hirofumi Tsumura[1]

  • [1] Department of Mathematics and Information Sciences Tokyo Metropolitan University Minami-Ohsawa, Hachioji 192-0397 Tokyo, Japan

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 1, page 219-226
  • ISSN: 1246-7405

Abstract

top
In this paper, we prove multiple analogues of famous Ramanujan’s formulas for certain Dirichlet series which were introduced in his well-known notebooks. Furthermore, we prove some multiple versions of analogous formulas of Ramanujan which were given by Berndt and so on.

How to cite

top

Tsumura, Hirofumi. "On multiple analogues of Ramanujan’s formulas for certain Dirichlet series." Journal de Théorie des Nombres de Bordeaux 20.1 (2008): 219-226. <http://eudml.org/doc/10830>.

@article{Tsumura2008,
abstract = {In this paper, we prove multiple analogues of famous Ramanujan’s formulas for certain Dirichlet series which were introduced in his well-known notebooks. Furthermore, we prove some multiple versions of analogous formulas of Ramanujan which were given by Berndt and so on.},
affiliation = {Department of Mathematics and Information Sciences Tokyo Metropolitan University Minami-Ohsawa, Hachioji 192-0397 Tokyo, Japan},
author = {Tsumura, Hirofumi},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {multiple zeta values},
language = {eng},
number = {1},
pages = {219-226},
publisher = {Université Bordeaux 1},
title = {On multiple analogues of Ramanujan’s formulas for certain Dirichlet series},
url = {http://eudml.org/doc/10830},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Tsumura, Hirofumi
TI - On multiple analogues of Ramanujan’s formulas for certain Dirichlet series
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 1
SP - 219
EP - 226
AB - In this paper, we prove multiple analogues of famous Ramanujan’s formulas for certain Dirichlet series which were introduced in his well-known notebooks. Furthermore, we prove some multiple versions of analogous formulas of Ramanujan which were given by Berndt and so on.
LA - eng
KW - multiple zeta values
UR - http://eudml.org/doc/10830
ER -

References

top
  1. B. C. Berndt, Generalized Dedekind eta-functions and generalized Dedekind sums. Trans. Amer. Math. Soc. 178 (1973), 495–508. Zbl0262.10015MR371817
  2. B. C. Berndt, Generalized Eisenstein series and modified Dedekind sums. J. Reine Angew. Math. 272 (1974), 182–193. Zbl0294.10018MR360471
  3. B. C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mountain J. Math. 7 (1977), 147–189. Zbl0365.10021MR429703
  4. 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. Zbl0384.10011MR514690
  5. B. C. Berndt, Ramanujan’s Notebooks, part II. Springer-Verlag, New-York, 1989. Zbl0716.11001
  6. B. C. Berndt, Ramanujan’s Notebooks, part V. Springer-Verlag, New-York, 1998. Zbl0886.11001
  7. K. Dilcher, Zeros of Bernoulli, generalized Bernoulli and Euler polynomials. Memoirs of Amer. Math. Soc. 386 (1988). Zbl0645.10015MR938890
  8. M. E. Hoffman, Multiple harmonic series. Pacific J. Math. 152 (1992), 275–290. Zbl0763.11037MR1141796
  9. K. Katayama, On Ramanujan’s formula for values of Riemann zeta-function at positive odd integers. Acta Arith. 22 (1973), 149–155. Zbl0222.10040
  10. M. Lerch, Sur la fonction ζ ( s ) pour valeurs impaires de l’argument. J. Sci. Math. Astron. pub. pelo Dr. F. Gomes Teixeira, Coimbra 14 (1901), 65–69. 
  11. S. L. Malurkar, On the application of Herr Mellin’s integrals to some series. J. Indian Math. Soc. 16 (1925/1926), 130–138. Zbl51.0267.06
  12. K. Matsumoto, H. Tsumura, A new method of producing functional relations among multiple zeta-functions. Quart. J. Math. 59 (2008), 55–83. Zbl1151.11045MR2392501
  13. D. Zagier, Values of zeta functions and their applications. In “Proc. First Congress of Math., Paris”, vol. II, Progress in Math. 120, Birkhäuser, 1994, 497–512. Zbl0822.11001

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.