Nielsen-Ramanujan Constants

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

			(3)
			(4)
			(5)
			(6)

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.