Relation on the zeta function
- From: "Jon Slaughter" <Jon_Slaughter@xxxxxxxxxxx>
- Date: Sat, 30 Apr 2005 21:16:15 -0500
I thought I would post this equation as it is quite interesting.
Int((f(t)-f(0))/(exp(t) - 1),t=0..infinity) =
Sum(f^(k)(0)*Zeta(k+1),k=1..infinity);
Now, this is very simple "formula" that really seems to be just a notational
"ruse", but might provide a new way of looking at the laplace
transform(Because that is bassicaly what it is).
Anyways, the point is by possibly chosing an appropriate f, one can get some
"nice" identities on zeta.
for example, one can choose f(t) = sin(t)
and one gets
1/2Picoth(Pi) - 1/2 = sum((-1)^(k+1)*Zeta(2*k),k=1..infinity)
Then one can choose f(t) = sin(xt) and get a relation with a free parameter
1/2Picoth(Pi*x) - 1/(2x) = sum((-1)^(k+1)*x^(2*k-1)*Zeta(2*k),k=1..infinity)
One can get relations for any function that has a known formula for its kth
derivative. (the point of the -f(0) in the integrand is to remove Zeta(1)
from the sum and hence to allow the integral to converge)
Anyways, I've used this for formulas on cos, exp and Bessel and got some
relationships that involve odd Zeta(Zeta at odd pos integral values) but
generally it seems if the sum contains an odd Zeta then the integral has no
known anylitic antiderivative or value.
It seems to me that Zeta(odd) shows up in just about everywhere(not sure if
naturally or by force). What I was thinking is that maybe the reason that
Zeta(odd) is so difficult to find is because there is incomplete knowledge
of the structure(and this knowlege may or may not be possible to get). What
I'm saying is that since Zeta(odd) as Zeta(even) shows up in so many places
that surely if Zeta(even) was so easy to get then Zeta(odd) shouldn't be
that hard(unless ofcourse there is something strange going on).
Another possibility I was thinking of is that Zeta(odd) is only
representable anylitically in terms of a constant that is not defined yet
and that is the reason we cannot get a "nice" formula for it(and everyone
things it should be preresented in Pi). Maybe Zeta(2n+1) = a_n*?^(2*n+1)
where ? is some constant that is not anylitically related to any of the
"standard" constants(But maybe related to some obscure constants possibly).
Anyways, one can use this formula to derive a lot of "useless" formulas that
seem just to be symbolic disguises for the same concept... but isn't that
what all identities are? Just different perspectives on the same abstract
idea.
Jon
.
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