Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T08:04:24.936Z Has data issue: false hasContentIssue false
  • English
  • Français

Two Dirichlet series evaluations found on page 196 of Ramanujan's Lost Notebook

Published online by Cambridge University Press:  28 February 2012

BRUCE C. BERNDT ,
HENG HUAT CHAN  and
YOSHIO TANIGAWA
BRUCE C. BERNDT
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, U.S.A. e-mail: berndt@illinois.edu
HENG HUAT CHAN
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore. e-mail: matchh@nus.edu.sg
YOSHIO TANIGAWA
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan. e-mail: tanigawa@math.nagoya-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

On page 196 in his lost notebook, S. Ramanujan offers evaluations of two particular Dirichlet series. In this paper, we establish Ramanujan's evaluations and more general results by various approaches. The different evaluations arising from different methods yield intriguing, unsuspecting identities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 2 , September 2012 , pp. 341 - 360
Copyright
Copyright © Cambridge Philosophical Society 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Apostol, T. M.Introduction to Analytic Number Theory (Springer-Verlag, 1986).Google Scholar
[2]Andrews, G. E. and Berndt, B. C.Ramanujan's Lost Notebook, Part IV (Springer, to appear).Google Scholar
[3]Berndt, B. C.Ramanujan's Notebooks, Part I (Springer–Verlag, 1985).CrossRefGoogle Scholar
[4]Berndt, B. C., Evans, R. J., and Williams, K. S.Gauss and Jacobi Sums (John Wiley, 1998).Google Scholar
[5]Berndt, B. C. and Schoenfeld, L.Periodic analogues of the Euler–Maclaurin and Poisson summation formulas with applications to number theory. Acta Arith. 28 (1975), 23–68.CrossRefGoogle Scholar
[6]Davenport, H.Multiplicative Number Theory. 3rd ed. (Springer, 2000).Google Scholar
[7]Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W.NIST Handbook of Mathematical Functions (Cambridge University Press, 2010).Google Scholar
[8]Ramanujan, S.The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi, 1988.Google Scholar
[9]Riordan, J.Introduction to Combinatorial Analysis (Dover, 2002).Google Scholar
[10]Riordan, J.Combinatorial Identities (Robert E. Krieger Publishing Company, 1979).Google Scholar