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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
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

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
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

[1]Apostol, T. M.Introduction to Analytic Number Theory (Springer-Verlag, 1986).Google Scholar
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Two Dirichlet series evaluations found on page 196 of Ramanujan's Lost Notebook
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