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February 4th, 2015, 09:52 AM  #1 
Senior Member Joined: Aug 2014 From: United States Posts: 137 Thanks: 21 Math Focus: Learning  Inverting the Riemann Zeta Function with the Mobius Function
I know that if $F(n)=\displaystyle\sum\limits_{dn} f(d)$, then $\displaystyle f(n)=\sum\limits_{dn} \mu(d) F(n/d)$ 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 dn part by letting n or n! approach infinity (or something of the sense), then maybe that way dn 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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function, inverting, mobius, riemann, zeta 
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