Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

In 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.

share|improve this question

1 Answer 1

up vote 8 down vote accepted

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.