Manipulating the Zeta Function

[MisterPants]

Already posted in OT:

I'm screwing something up when I manipulate Euler's Zeta Function. Somebody tell me what it is. I'll be using Z for zeta(s), and when i type 1/n assume i mean 1/n^s where s >1.

A better version:
Z=1+1/2+1/3...
Z/2=1/2+1/4+1/6...
Z-Z/2=1+1/3+1/5+1/7...
Z-Z/2=Z(1-1/2)=Z/2

Thus:
1+1/3+1/5+1/7...=1/2+1/4+1/6+1/8...

1/3(Z-Z/2)=1/3+1/9+1/15...
Z-Z/2-(Z-Z/2)/3=1+1/5+1/7+1/11...

Can somebody show me how to get 1/Pi(1-p^-s)=Sigma(n^-s)

(Pi being the multiplicative analog of Sigma, n being the natural numbers, and p being the primes)

 

[fell8]

My head hurts.

 

[Mday]

this has to be the worst notation i have ever seen used for ascii math.

the level of math required to understand is within my level of education, but i have never encountered this as my relationship with math prevents it.

http://mathworld.wolfram.com/RiemannZetaFunction.html

enjoy the read with proper notation.

 

[MisterPants]

For better notation, go here: http://www.wikipedia.org/wiki/Riemann_zeta_function

The mathworld article doesn't deal much with the non-complex zeta function.

 

[Jeff7]

Originally posted by: fell8
My head hurts.

I'm having flashbacks to Calculus in high school. I didn't pass the final (think I got a 68% on it) but I did pass the class. And I was usually a high honor roll student; this kind of math just did not sit well with me.

 

[ermular]

Originally posted by: Jeff7

Originally posted by: fell8
My head hurts.

I'm having flashbacks to Calculus in high school. I didn't pass the final (think I got a 68% on it) but I did pass the class. And I was usually a high honor roll student; this kind of math just did not sit well with me.

This is well beyond high school calculus