In [Are the nontrivial zeros of the Riemann zeta simple?](https://mathoverflow.net/questions/59770/are-the-non-trivial-zeros-of-zeta-simple), I asked whether it was known that all non-trivial zeros of the Riemann Zeta function were simple or not. It appears that such a proof is missing. But are there partial results concerning a possible upper bound for the multiplicity of any non-trivial zero of Zeta? I'd be interested even in rather weak results like $m\_{\rho}=O(f(T\_{\rho}))$, with $\rho$ a non trivial zero of Zeta, $T\_{\rho}$ its imaginary part and $f$ a map such that $f(x)=o(x)$.  

Thanks in advance.



Do we know an upper bound for the multiplicity of the non-trivial zeros of Zeta?
It is known that the number of zeros with $T-1 < Im(\rho)<T+1$ is $O(\log(T))$. Therefore
the multiplicity of a zero $\beta+\gamma i$ will be less than $C \log|\gamma|$ for some
absolute constant $C$.


(The result quoted is Theorem 9.2 in Titchmarsh's book.)


There are other theorems that say that a proportion of zeros are simple.


