More on the zeta function....

• Mar 1st 2010, 11:01 AM
bigdoggy
More on the zeta function....
I cannot see why the folowing is correct:
$\displaystyle \zeta(s)^{-1}= \Sigma {\mu(n) \over n^s}$
where $\displaystyle \mu$ is the Mobius function....
• Mar 1st 2010, 12:12 PM
NonCommAlg
Quote:

Originally Posted by bigdoggy
I cannot see why the folowing is correct:
$\displaystyle \zeta(s)^{-1}= \Sigma {\mu(n) \over n^s}$
where $\displaystyle \mu$ is the Mobius function....

well, by the Euler product formula or just a simple observation we have: $\displaystyle \zeta(s)=\prod_p \left(1+ \frac{1}{p^s} + \frac{1}{p^{2s}} + \cdots \right)=\prod_p \frac{1}{(1- \frac{1}{p^s})},$ where the product is over all (distinct) primes.

thus $\displaystyle \frac{1}{\zeta(s)}=\prod_p \left(1-\frac{1}{p^s} \right)=\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s},$ by the definition of $\displaystyle \mu(n)$ and after expanding the product.
• Mar 1st 2010, 12:15 PM
aliceinwonderland
Quote:

Originally Posted by bigdoggy
I cannot see why the folowing is correct:
$\displaystyle \zeta(s)^{-1}= \Sigma {\mu(n) \over n^s}$
where $\displaystyle \mu$ is the Mobius function....

If s > 1,

$\displaystyle \sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\prod_p(1-p^{-s})=\frac{1}{\zeta(s)}.$