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:

“I look on trade and every mechanical craft as education also. But let me discriminate what is precious herein. There is in each of these works an act of invention, an intellectual step, or short series of steps taken; that act or step is the spiritual act; all the rest is mere repetition of the same a thousand times.”
—Ralph Waldo Emerson (1803–1882)

“Preschoolers think and talk in concrete, literal terms. When they hear a phrase such as “losing your temper,” they may wonder where the lost temper can be found. Other expressions they may hear in times of crisis—raising your voice, crying your eyes out, going to pieces, falling apart, picking on each other, you follow in your father’s footsteps—may be perplexing.”
—Ruth Formanek (20th century)