## Nielsen-Ramanujan Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

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 V. Adamchik (Finch)
 (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.

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/nielram/nielram.html

Flajolet, P. and Salvy, B. Euler Sums and Contour Integral Representation.'' Submitted to Experim. Math 1997. http://pauillac.inria.fr/algo/flajolet/Publications/publist.html.