My Math Forum Inverting the Riemann Zeta Function with the Mobius Function

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 February 4th, 2015, 10:52 AM #1 Senior Member   Joined: Aug 2014 From: United States Posts: 136 Thanks: 21 Math Focus: Learning 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.

 Tags function, inverting, mobius, riemann, zeta

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