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Nielsen-Ramanujan Constants

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N. Nielsen (1909) and Ramanujan (Berndt 1985) considered the integrals

a_k=int_1^2((lnx)^k)/(x-1)dx.
(1)

They found the values for k=1 and 2. The general constants for k>3 were found by Levin (1950) and, much later, independently by V. Adamchik (Finch 2003),

a_p=p!zeta(p+1)-(p(ln2)^(p+1))/(p+1)-p!sum_(k=0)^(p-1)(Li_(p+1-k)(1/2)(ln2)^k)/(k!),
(2)

where zeta(z) is the Riemann zeta function and Li_n(x) is the polylogarithm. The first few values are

a_1=1/2zeta(2)=1/(12)pi^2
(3)
a_2=1/4zeta(3)
(4)
a_3=1/(15)pi^4+1/4pi^2(ln2)^2-1/4(ln2)^4-6Li_4(1/2)-(21)/4(ln2)zeta(3)
(5)
a_4=2/3pi^2(ln2)^3-4/5(ln2)^5-24(ln2)Li_4(1/2)-24Li_5(1/2)-(21)/2(ln2)^2zeta(3)+24zeta(5).
(6)

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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Nielsen-Ramanujan Constants

Cite this as:

Weisstein, Eric W. "Nielsen-Ramanujan Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Nielsen-RamanujanConstants.html

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