Khinchin's Constant - Series Expressions

Series Expressions

Khinchin's constant may be expressed as a rational zeta series in the form

\log K_0 = \frac{1}{\log 2} \sum_{n=1}^\infty
\frac {\zeta (2n)-1}{n} \sum_{k=1}^{2n-1} \frac{(-1)^{k+1}}{k}

or, by peeling off terms in the series,

\log K_0 = \frac{1}{\log 2} \left[
\sum_{k=3}^N \log \left(\frac{k-1}{k} \right) \log \left(\frac{k+1}{k} \right)
+ \sum_{n=1}^\infty
\frac {\zeta (2n,N)}{n} \sum_{k=1}^{2n-1} \frac{(-1)^{k+1}}{k}
\right]

where N is an integer, held fixed, and ζ(s, n) is the Hurwitz zeta function. Both series are strongly convergent, as ζ(n) − 1 approaches zero quickly for large n. An expansion may also be given in terms of the dilogarithm:

\log K_0 = \log 2 + \frac{1}{\log 2} \left[
\mbox{Li}_2 \left( \frac{-1}{2} \right) +
\frac{1}{2}\sum_{k=2}^\infty (-1)^k \mbox{Li}_2 \left( \frac{4}{k^2} \right)
\right].

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