Thanks, Sandwich, although I don't really mind it too much.

Some of us less enlightened (but no less interested in mathematics) haven't heard of some of these things. Links are A-1 SUPAR.
For example, what is G, the gamma function, the beta function, and Khinchin's Law?
On that note, thanks for describing K(x).
What is it used for?
Sure; I don't have much time at the moment, but I will point you to some links.
The gamma function is basically an extension of the usual factorial to noninteger arguments, and the two are related by
G(x+1) = x! . There is a bunch of information about this one available here:
http://mathworld.wolfram.com/GammaFunction.htmlThere exist several formulas and identities for this function as well, some of which can be used to prove that product result I gave earlier.
G is called Glaisher's constant and is actually defined by that limit; I think it is not known whether the constant is irrational, (although it is likely to be). It can be written in terms of derivatives of the zeta function, but not in terms of any of the usual functions or constants. I would post a link for this as well but the site I usually go to for constant-related information is currently down.
Here's one neat identity I can remember at the moment:

(pi and e you know, G is Glaisher's constant, and
g is Euler's constant)
As for Khinchin's Law, it has to do with continued fractions. A CF is an "infinite fraction" of the following form:

Just like there is a decimal representation for every number, there is also a CF representation. For rational numbers, the CF does not go on forever but terminates. It turns out that, while irrational numbers have no periodic repeats in their decimal form, such patterns do frequently exist in CF form, and many common roots and constants can be expressed as CFs with some pattern in them. What Khinchin found is that, for almost all real numbers, taking the geometric mean of the numbers in the CF (a
1, a
2, a
3, etc.) actually yields a special constant that is
independent of the original number. In fact, if you select a real number completely randomly, the probability that it will satisfy Khinchin's law is actually 100%. (there are numbers that do not satisfy it, but there are infinitely more that do satisfy it) The constant is called Khinchin's constant and is given by the following infinite product:

It is not known to be expressible in terms of any other functions or constants, although there are a few really messy integrals and sums that give it.
I am not too experienced with this stuff myself so I do not know exactly why this law is true, but you can find some more information here:
http://mathworld.wolfram.com/KhinchinsConstant.htmlIf you find this stuff interesting, let me know, as I can go on about this all day.

Are you sure you have the series correct?

looks like I wrote that one in all wrong; it should be ( e
p - e
-p ) / (2
p), or sinh(
p)/
p. sorry about that; I think those errors got buried in all the html/vb tags.
