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February 4th, 2015, 09:52 AM   #1
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Inverting the Riemann Zeta Function with the Mobius Function

I know that if $F(n)=\displaystyle\sum\limits_{d|n} f(d)$, then $\displaystyle f(n)=\sum\limits_{d|n} \mu(d) F(n/d)$

How do we use this to find the reciprocal of the Riemann Zeta function??

$\displaystyle\frac{1} {\zeta(s)}=\sum\limits_{n=1}^\infty \frac{\mu(n)} {n^s}$

And is there a way to completely invert the Riemann zeta function?

I think maybe we can get the d|n part by letting n or n! approach infinity (or something of the sense), then maybe that way d|n would allow us to use $d\in \mathbb{N}$

But I am interested as to what methods are used to derive the reciprocal of the zeta function.
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