N. Nielsen (1909) and Ramanujan (Berndt 1985) considered the integrals
 |
(1) |
They found the values for 
and 2. The general constants for 
were found by Levin (1950) and, much later, independently
by V. Adamchik (Finch 2003),
 |
(2) |
where 
is the Riemann zeta function and 
is the polylogarithm.
The first few values are
See also
Polylogarithm, Riemann Zeta Function
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References
Berndt, B. C. Ramanujan's Notebooks, Part I. New York: Springer-Verlag, 1985.Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; and Lisonek, P. "Special Values of Multidimensional Polylogarithms." Trans. Amer. Math. Soc. 353, 907-941, 2001.Finch, S. R. "Apéry's Constant: Polylogarithms." §1.6.8 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 47-48, 2003.Flajolet, P. and Salvy, B. "Euler Sums and Contour Integral Representation." Experim. Math. 7, 15-35, 1998.Levin, V. I. "About a Problem of S. Ramanujan." Uspekhi Mat. Nauk 5, 161-166, 1950.Nielsen, N. "Der Eulersche Dilogarithmus und seine Verallgemeinerungen." Nova Acta Leopoldina, Abh. der Kaiserlich Leopoldinisch-Carolinischen Deutschen Akad. der Naturforsch. 90, 121-212, 1909.
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Cite this as:
Weisstein, Eric W. "Nielsen-Ramanujan Constants." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Nielsen-RamanujanConstants.html