# gamma function

• Nov 30th 2009, 10:18 PM
dymin3
gamma function
let Pn denote the nth prime (P1=2, P2=3, etc) If Re(s) > 1, show that ∑(from n=1 to infinite) 1/n^s = Π(from n=1 to infinite) 1/(1-Pn^-s).

I was thinking to expand each factor z/(1-Pn^-s) in a geometric series and use unique factorization of integers into a product of primes but i was not sure...

plz help me!! (Crying)
• Dec 1st 2009, 10:48 AM
tonio
Quote:

Originally Posted by dymin3
let Pn denote the nth prime (P1=2, P2=3, etc) If Re(s) > 1, show that ∑(from n=1 to infinite) 1/n^s = Π(from n=1 to infinite) 1/(1-Pn^-s).

I was thinking to expand each factor z/(1-Pn^-s) in a geometric series and use unique factorization of integers into a product of primes but i was not sure...

plz help me!! (Crying)

I'll show you some highlights of what Euler did in his time since it is a delightful thing, though this is NOT a formal proof:

(1) $\displaystyle \zeta(s)=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s} +\ldots\,\Longrightarrow \frac{1}{2^s}\zeta(s)=\frac{1}{2^s}+\frac{1}{4^s}+ \frac{1}{6^s}+\ldots$ , and substracting this from (1):

(2) $\displaystyle \left(1-\frac{1}{2^s}\right)\zeta(s)$ $\displaystyle =\frac{1}{1^s}+\frac{1}{3^s}+\frac{1}{5^s}+\ldots$

$\displaystyle \frac{1}{3^s}\left(1-\frac{1}{2^s}\right)\zeta(s)=\frac{1}{3^s}+\frac{1 }{6^s}+\frac{1}{9^s}+\ldots$ , and substracting this from (2) above:

(3) $\displaystyle \left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{2^s}\right)\zeta(s)=\frac{1}{1^s}+\frac{1 }{5^s}+\frac{1}{7^s}+\frac{1}{11^s}+\ldots$

Continue in this fashion and you'll get

$\displaystyle \ldots\left(1-\frac{1}{13^s}\right)\left(1-\frac{1}{11^s}\right)\left(1-\frac{1}{7^s}\right)\left(1-\frac{1}{5^s}\right)\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{2^s}\right)\zeta(s)=1$

Well, now just solve for $\displaystyle \zeta(s)$ above...(Happy)

Tonio

Ps If after seeing the above one doesn't crash down in deep love with mathematics then all is lost...(Crying)

Ps of Ps BTW, what does the gamma function has to do with all this??
• Dec 1st 2009, 12:29 PM
dymin3
Thank you so much!! oh as for the title, originally i had two question but one got deleted so... and that one was about gamma function