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

$$a_k=\int_1^2 {(\ln x)^k\over x-1}\,dx.$$ (1)

They found the values for $k=1$ and 2. The general constants for $k>3$ were found by V. Adamchik (Finch)

$$a_p=p!\zeta(p+1)-{p(\ln 2)^{p+1}\over p+1}-p! \sum_{k=0}^{p-... ...}\nolimits _{p+1-k}({\textstyle{1\over 2}})(\ln 2)^k\over k!},$$ (2)

where $\zeta(z)$ is the Riemann Zeta Function and $\mathop{\rm Li}\nolimits _n(x)$ is the Polylogarithm. The first few values are

$$a_1 = {\textstyle{1\over 2}}\zeta(2)={\textstyle{1\over 12}}\pi^2$$ (3)

$$a_2 = {\textstyle{1\over 4}}\zeta(3)$$ (4)

$$a_3 = {\textstyle{1\over 15}}\pi^4+{\textstyle{1\over 4}}\pi^2(\ln 2)^2... ...rm Li}\nolimits _4({\textstyle{1\over 2}})-{\textstyle{21\over 4}}\ln 2\zeta(3)$$ (5)

$$a_4 = {\textstyle{2\over 3}}\pi^2(\ln 2)^3-{\textstyle{4\over 5}}(\ln 2... ..._5({\textstyle{1\over 2}})-{\textstyle{21\over 2}}(\ln 2)^2\zeta(3)+24\zeta(5).$$ (6)

See also Polylogarithm, Riemann Zeta Function

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.