I saw a statement about the Riemann hypothesis in Wikipedia, stating the following:
$\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=\frac{1}{\zeta(s)}$ holds for $Re(s)>\frac{1}{2}$ is equivalent to the Riemann hypothesis.
I tried to find a reference about it but I can't find it. May anyone provide a reference to the statement(with a proof) to it?
Best Answer
The paper Riemann hypothesis equivalences, Robin inequality, Lagarias criterion, and Riemann hypothesis lists $36$ equivalent statements to RH, with references to complete proofs. We have the following statement on page $8$:
Equivalence 8: RH is equivalent to the fact that the following series converges for $ℜ(s) > \frac{1}{2}$, see [48]: $$ \sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}. $$ And $[48]$ is the book by Titchmarsh, The theory of the Riemann zeta function,second ed., Clarendon press. Oxford University press, New York, 1986.
There are some more posts at MO and MSE, e.g. this one, where it is also referenced.