Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same?

Question

All:

Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same ?

They shall be all different, right ? Is there a proof of this statement ?

Thank you.

Answer 1

At the moment, this is completely unknown. If $\zeta(s) = \zeta'(s) = 0$, then $s$ is (at least) a double zero. We do not know whether $\zeta(s)$ should have multiple-zeroes or not. Further, multiple zeroes are somewhat independent of the Riemann Hypothesis. There doesn't seem to be any particular reason in particular for them to, or not to, occur.

In terms of what is currently known, in 1974 Montgomery showed that the Riemann Hypothesis implies that about 67 percent of zeroes of the Riemann zeta function are simple zeroes. [Montgomery, Distribution of zeros of the Riemann zeta function, Proc. Internat. Congr. Math. Vancouver, 1974, pp. 379-381].

In much greater generality, one might consider the same question for general $L$-functions. In short, many $L$-functions do have zeroes of higher multiplicity. The Birch and Swinnerton-Dyer Conjecture indicates that the multiplicity of a particular zero of an $L$-function relates to the rank of an associated elliptic curve, for instance. Some Dedekind zeta functions over number fields also have multiple zeroes.