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:

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

Read more about this topic: Khinchin's Constant

Famous quotes containing the words series and/or expressions:

“The woman’s world ... is shown as a series of limited spaces, with the woman struggling to get free of them. The struggle is what the film is about; what is struggled against is the limited space itself. Consequently, to make its point, the film has to deny itself and suggest it was the struggle that was wrong, not the space.”
—Jeanine Basinger (b. 1936)

“Compare the history of the novel to that of rock ‘n’ roll. Both started out a minority taste, became a mass taste, and then splintered into several subgenres. Both have been the typical cultural expressions of classes and epochs. Both started out aggressively fighting for their share of attention, novels attacking the drama, the tract, and the poem, rock attacking jazz and pop and rolling over classical music.”
—W. T. Lhamon, U.S. educator, critic. “Material Differences,” Deliberate Speed: The Origins of a Cultural Style in the American 1950s, Smithsonian (1990)