Abstract
In this chapter, we relate two well-known identities, with the names of N.S. Koshliakov and A.P. Guinand attached to them, which were proved by Ramanujan and recorded in his lost notebook before their discoveries by the aforementioned mathematicians. Ramanujan also derived some related formulas that have not been rediscovered by others.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part II, Springer, New York, 2009.
P.T. Bateman and E. Grosswald, On Epstein’s zeta function, Acta Arith. 9 (1964), 365–373.
B.C. Berndt, Identities involving the coefficients of a class of Dirichlet series. III., Trans. Amer. Math. Soc. 146 (1969), 323–348.
B.C. Berndt, Identities involving the coefficients of a class of Dirichlet series. V., Trans. Amer. Math. Soc. 160 (1971), 139–156.
B.C. Berndt, Periodic Bernoulli numbers, summmation formulas and applications, in Theory and Application of Special Functions, R.A. Askey, ed., Academic Press, New York, 1975, pp. 143–189.
B.C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989.
B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.
B.C. Berndt, An unpublished manuscript of Ramanujan on infinite series identities, J. Ramanujan Math. Soc. 19 (2004), 57–74.
B.C. Berndt, A. Dixit, and J. Sohn, Character analogues of theorems of Ramanujan, Koshliakov, and Guinand, Adv. Appl. Math. 46 (2011), 54–70.
B.C. Berndt, Y. Lee, and J. Sohn, Koshliakov’s formula and Guinand’s formula in Ramanujan’s lost notebook, in Surveys in Number Theory, K. Alladi, ed., Springer, New York, 2008, pp. 21–42.
B.C. Berndt and A.J. Yee, Ramanujan’s contributions to Eisenstein series, especially in his lost notebook, in Number Theoretic Methods – Future Trends, C. Jia and S. Kanemitsu, eds., Kluwer, Dordrecht, 2002, pp. 31–53; abridged version, A survey on Eisenstein series in Ramanujan’s lost notebook, in New Aspects of Analytic Number Theory, Y. Tanigawa, ed., Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 2002, pp. 130–141.
H. Cohen, Some formulas of Ramanujan involving Bessel functions, Publ. Math. Besancon (2010), 59–68.
A. Dixit, Series transformations and integrals involving the Riemann Ξ-function, J. Math. Anal. Appl. 368 (2010), 358–373.
A. Dixit, Transformation formulas associated with integrals involving the Riemann Ξ-function, Monatsh. für Math. 164 (2011), 133–156.
A. Dixit, Character analogues of Ramanujan type integrals involving the Riemann Ξ-function, Pacific J. Math. 255 (2012), 317–348.
A.L. Dixon and W.L. Ferrar, On the summation formulae of Voronoı̈ and Poisson, Quart. J. Math. (Oxford) 8 (1937), 66–74.
W.L. Ferrar, Some solutions of the equation \(F(t) = F({t}^{-1})\), J. London Math. Soc. 11 (1936), 99–103.
I.S. Gradshteyn and I.M. Ryzhik, eds., Table of Integrals, Series, and Products, 5th ed., Academic Press, San Diego, 1994.
A.P. Guinand, Some rapidly convergent series for the Riemann ξ-function, Quart. J. Math. (Oxford) 6 (1955), 156–160.
S. Kanemitsu, Y. Tanigawa, H. Tsukada, and M. Yoshimoto, On Bessel series expressions for some lattice sums: II, J. Physics A: Mathematics and General, 37 (2004), 719–734.
S. Kanemitsu, Y. Tanigawa, and M. Yoshimoto, On rapidly convergent series for the Riemann zeta-values via the modular relation, Abh. Math. Sem. Univ. Hamburg 72 (2002), 187–206.
H. Kober, Transformationsformeln gewisser Besselscher Reihen Beziehungen zu Zeta-functionen, Math. Z. 39 (1934), 609–624.
N.S. Koshliakov, On Voronoı̈’s sum-formula, Mess. Math. 58 (1929), 30–32.
N.S. Koshliakov, Some integral representations of the square of Riemann’s function Ξ(t), Dokl. Akad. Nauk. 2 (1934), 401–405.
T. Kubota, Elementary Theory of Eisenstein Series, Kodansha, Tokyo, 1973.
H. Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen and die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141–183.
C.J. Moreno, Advanced Analytic Number Theory: L-Functions, Math. Surveys and Monographs, Vol. 115, American Mathematical Society, Providence, RI, 2005.
F. Oberhettinger and K.L. Soni, On some relations which are equivalent to functional equations involving the Riemann zeta function, Math. Z. 127 (1972), 17–34.
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; second ed, 2012.
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
A. Selberg, Collected Papers, Vol. I, Springer-Verlag, Berlin, 1989.
A. Selberg and S. Chowla, On Epstein’s zeta-function (I), Proc. Nat. Acad. Sci. (USA) 35 (1949), 371–374.
A. Selberg and S. Chowla, On Epstein’s zeta-function, J. Reine Angew. Math. 227 (1967), 86–110.
K. Soni, Some relations associated with an extension of Koshliakov’s formula, Proc. Amer. Math. Soc. 17 (1966), 543–551.
A. Terras, Harmonic Analysis on Symmetric Spaces and Applications I, Springer-Verlag, New York, 1985.
E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford, 1951.
M.G. Voronoı̈, Sur une fonction transcendante et ses applications à la sommation de quelques séries, Ann. École Norm. Sup. (3) 21 (1904), 207–267, 459–533.
G.N. Watson, Some self-reciprocal functions, Quart. J. Math. (Oxford) 2 (1931), 298–309.
G.N. Watson, Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1966.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Andrews, G.E., Berndt, B.C. (2013). Koshliakov’s Formula and Guinand’s Formula. In: Ramanujan's Lost Notebook. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4081-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4081-9_3
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4080-2
Online ISBN: 978-1-4614-4081-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)