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

  • BRUCE C. BERNDT (a1), HENG HUAT CHAN (a2) and YOSHIO TANIGAWA (a3)

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.

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References

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[1]Apostol, T. M.Introduction to Analytic Number Theory (Springer-Verlag, 1986).
[2]Andrews, G. E. and Berndt, B. C.Ramanujan's Lost Notebook, Part IV (Springer, to appear).
[3]Berndt, B. C.Ramanujan's Notebooks, Part I (Springer–Verlag, 1985).
[4]Berndt, B. C., Evans, R. J., and Williams, K. S.Gauss and Jacobi Sums (John Wiley, 1998).
[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), 2368.
[6]Davenport, H.Multiplicative Number Theory. 3rd ed. (Springer, 2000).
[7]Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W.NIST Handbook of Mathematical Functions (Cambridge University Press, 2010).
[8]Ramanujan, S.The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi, 1988.
[9]Riordan, J.Introduction to Combinatorial Analysis (Dover, 2002).
[10]Riordan, J.Combinatorial Identities (Robert E. Krieger Publishing Company, 1979).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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