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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.
\end{displaymath} (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!},
\end{displaymath} (2)

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

$\displaystyle a_1$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\zeta(2)={\textstyle{1\over 12}}\pi^2$ (3)
$\displaystyle a_2$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}\zeta(3)$ (4)
$\displaystyle a_3$ $\textstyle =$ $\displaystyle {\textstyle{1\over 15}}\pi^4+{\textstyle{1\over 4}}\pi^2(\ln 2)^2...
...rm Li}\nolimits _4({\textstyle{1\over 2}})-{\textstyle{21\over 4}}\ln 2\zeta(3)$ (5)
$\displaystyle a_4$ $\textstyle =$ $\displaystyle {\textstyle{2\over 3}}\pi^2(\ln 2)^3-{\textstyle{4\over 5}}(\ln 2...
..._5({\textstyle{1\over 2}})-{\textstyle{21\over 2}}(\ln 2)^2\zeta(3)+24\zeta(5).$ (6)

See also Polylogarithm, Riemann Zeta Function


Berndt, B. C. Ramanujan's Notebooks, Part I. New York: Springer-Verlag, 1985.

Finch, S. ``Favorite Mathematical Constants.''

Flajolet, P. and Salvy, B. ``Euler Sums and Contour Integral Representation.'' Submitted to Experim. Math 1997.

© 1996-9 Eric W. Weisstein