Asymptotic relation to a function

**mathematicaphoenix** · January 1st 2012, 07:54 AM

Given the function,
$\zeta (M,s)=\zeta (x)\prod_{p\leq M}(1-\frac{1}{p^{s}})$
where the product is taken over primes $p$ .
How do you find the asymptotic relation
$\ln \zeta (7,s)\sim \frac{1}{11^{s}}$

Thanks

**chisigma** · January 1st 2012, 09:09 AM

Originally Posted by mathematicaphoenix

Given the function,
$\zeta (M,s)=\zeta (x)\prod_{p\leq M}(1-\frac{1}{p^{s}})$
where the product is taken over primes $p$ .
How do you find the asymptotic relation
$\ln \zeta (7,s)\sim \frac{1}{11^{s}}$

Thanks

By definition is...

$\zeta(M,s)= \zeta(s)\ \prod_{p \le M} (1-\frac{1}{p^{s}}) = \frac{\prod_{p \le M} (1-\frac{1}{p^{s}})}{\prod_{p} (1-\frac{1}{p^{s}})}= \frac{1}{\prod_{p >M} (1-\frac{1}{p^{s}})}$ (1)

Proceeding as in...

http://www.mathhelpforum.com/math-he...on-194803.html

... You find...

$\ln \zeta(M,s)= - \sum_ {p>M} \ln (1-\frac{1}{p^{s}}) =$

$= \sum_{p>M} (\frac{1}{p^{s}} + \frac{1}{2\ p^{2 s}} + \frac{1}{3\ p^{3 s}} +...)$ (2)

For M=7 You obtain as in the post I indicated ...

$\lim_{s \rightarrow + \infty} 11^{s}\ \ln \zeta(7,s) = 1$ (3)

... that can be written as...

$\ln \zeta(7,s) \sim \frac{1}{11^{s}}$ (4)

Kind regards

$\chi$ $\sigma$

**mathematicaphoenix** · January 1st 2012, 03:32 PM

Thanks. Actually I tried first as in the other post. But applied laws of logarithm first an couldn't get (1).
Anyway I got it now. Thanks a lot once again.

Regards
Nabigh

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