Editing Khinchin's constant (section)

==Series expressions==
Khinchin's constant can be given by the following infinite product:

:<math>K_0=\prod_{r=1}^\infty {\left( 1+{1\over r(r+2)}\right)}^{\log_2 r}</math>

This implies:

:<math>\ln K_0=\sum_{r=1}^\infty \ln{\left( 1+{1\over r(r+2)}\right)}{\log_2 r}</math>

Khinchin's constant may also be expressed as a [[rational zeta series]] in the form<ref>Bailey, Borwein & Crandall, 1997. In that paper, a slightly non-standard definition is used for the Hurwitz zeta function.</ref>

:<math>\ln K_0 = \frac{1}{\ln 2} \sum_{n=1}^\infty 
\frac {\zeta (2n)-1}{n} \sum_{k=1}^{2n-1} \frac{(-1)^{k+1}}{k}
</math>
or, by peeling off terms in the series, 
:<math>\ln K_0 = \frac{1}{\ln 2} \left[
-\sum_{k=2}^N \ln \left(\frac{k-1}{k} \right) \ln \left(\frac{k+1}{k} \right)
+ \sum_{n=1}^\infty 
\frac {\zeta (2n,N+1)}{n} \sum_{k=1}^{2n-1} \frac{(-1)^{k+1}}{k}
\right]
</math>

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

:<math>\ln \frac{K_0}{2} = \frac{1}{\ln 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].
</math>