Summary
Two proofs are given for a series transformation formula involving the logarithmic derivative of the Gamma function found in Ramanujan’s lost notebook. The transformation formula is connected with a certain integral embodying the Riemann zeta function that is similar to integrals examined by Ramanujan in his one published paper on the zeta function.
In Memory of Alladi Ramakrishnan
Mathematics Subject Classification (2000) Primary, 11M06; Secondary, 33B15
Research partially supported by grant H98230-07-1-0088 from the National Security Agency.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The authors are indebted to M. L. Glasser for the proof of this lemma. The authors’ original proof of this lemma was substantially longer than Glasser’s given here.
References
M. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1965.
G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part IV, Springer, New York, (in press).
B.C. Berndt, Ramanujan’s quarterly reports, Bull. Lond. Math. Soc. 16 (1984), 449–489.
B.C. Berndt, Ramanujan’s Notebooks, Part I, Springer, New York, 1985.
J.B. Conway, Functions of One Complex Variable, 2nd ed., Springer, New York, 1978.
A. Dixit, Analogues of a transformation formula of Ramanujan, submitted for publication.
A. Dixit, Series transformations and integrals involving the Riemann Ξ-function, J. Math. Anal. Appl. 368 (2010), 358–373.
I.S. Gradshteyn and I.M. Ryzhik, eds., Table of Integrals, Series, and Products, 5th ed., Academic, San Diego, 1994.
A.P. Guinand, On Poisson’s summation formula, Ann. Math. (2) 42 (1941), 591–603.
A.P. Guinand, Some formulae for the Riemann zeta-function, J. Lond. Math. Soc. 22 (1947), 14–18.
A.P. Guinand, A note on the logarithmic derivative of the Gamma function, Edinb. Math. Notes 38 (1952), 1–4.
A.P. Guinand, Some finite identities connected with Poisson’s summation formula, Proc. Edinb. Math. Soc. (2) 12 (1960), 17–25.
G.H. Hardy, Note by G.H. Hardy on the preceding paper, Quart. J. Math. 46 (1915), 260–261.
S. Ramanujan, New expressions for Riemann’s functions ξ(s) and Ξ(s), Quart. J. Math. 46 (1915), 253–260.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927; reprinted by Chelsea, New York, 1962; reprinted by the American Mathematical Society, Providence, RI, 2000.
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer New York
About this chapter
Cite this chapter
Berndt, B.C., Dixit, A. (2010). A Transformation Formula Involving the Gamma and Riemann Zeta Functions in Ramanujan’s Lost Notebook. In: Alladi, K., Klauder, J., Rao, C. (eds) The Legacy of Alladi Ramakrishnan in the Mathematical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6263-8_12
Download citation
DOI: https://doi.org/10.1007/978-1-4419-6263-8_12
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-6262-1
Online ISBN: 978-1-4419-6263-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)