The Ramanujan Journal

, Volume 21, Issue 1, pp 99–122 | Cite as

Ramanujan summation and the exponential generating function \(\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta^{\prime}(-k)\)

  • B. Candelpergher
  • H. Gopalkrishna Gadiyar
  • R. PadmaEmail author


In the sixth chapter of his notebooks, Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions ∑n≥1log n e nz and \(\sum_{n\geq 1}H_{n}^{(j)}~e^{-nz}\) where \(H_{n}^{(j)}=\sum_{m=1}^{n}\frac{1}{m^{j}}\) . We find a surprising relation between the two sums when j=1 from which follows a formula that connects the derivatives of the Riemann zeta-function at the negative integers to the Ramanujan sum of the divergent Euler sums ∑n≥1 n k H n k ≥ 0, where \(H_{n}=H_{n}^{(1)}\) . Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.

Divergent series Euler sums Generating function Riemann zeta-function Borel sum Laplace transform 

Mathematics Subject Classification (2000)

11M06 65B15 40G99 


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  1. 1.
    Amdeberhan, T., Espinosa, O., Moll, V.H.: The Laplace transform of the digamma function: an integral due to Glasser, Manna and Oloa. Proc. Am. Math. Soc. 136, 3211–3221 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer International Student Edition. Springer, Berlin (1980) Google Scholar
  3. 3.
    Berndt, B.: Ramanujan’s Notebooks, Part I. Springer, Berlin (1985) zbMATHGoogle Scholar
  4. 4.
    Berndt, B.: Ramanujan’s Notebooks, Part II. Springer, Berlin (1985) Google Scholar
  5. 5.
    Boyadzhiev, K.N., Gadiyar, H.G., Padma, R.: A note on the values of an Euler sum at negative integers and relation to a convolution of Bernoulli numbers. Bull. Korean Math. Soc. 45(2), 277–283 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Candelpergher, B., Coppo, M.A., Delabaere, E.: La sommation de Ramanujan. Enseign. Math. 43, 93–132 (1997) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Flajolet, P., Salvy, B.: Euler sums and contour integral representations. Exp. Math. 7(1), 15–35 (1998) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Gadiyar, H.G., Padma, R.: A comment on Matiyasevich’s identity #0102 with Bernoulli numbers.
  9. 9.
    Gessel, I.M.: On Miki’s identities for Bernoulli numbers. J. Number Theory 110, 75–82 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Glasser, M.L., Manna, D.: On the Laplace transform of the Psi function. In: Amdeberhan, T., Moll, V. (eds.) Tapas in Experimental Mathematics. Contemporary Mathematics, vol. 457, pp. 193–202. AMS, Providence (2008) Google Scholar
  11. 11.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 5th edn. Academic Press, San Diego (1994), Alan Jeffrey (ed.) zbMATHGoogle Scholar
  12. 12.
    Hardy, G.H.: Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Chelsea, New York (1940) Google Scholar
  13. 13.
    Hardy, G.H.: Divergent Series. Clarendon, Oxford (1949) zbMATHGoogle Scholar
  14. 14.
    Roman, S.: The logarithmic binomial formula. Am. Math. Mon. 99(7), 641–648 (1992) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • B. Candelpergher
    • 1
  • H. Gopalkrishna Gadiyar
    • 2
  • R. Padma
    • 2
    Email author
  1. 1.Laboratory J.A. Dieudonné, UMRCNRS No. 6621University of Nice-Sophia AntipolisNice Cedex 2France
  2. 2.AU-KBC Research CentreChennaiIndia

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