Size of $\zeta'(s)$ at its zeros

Question

How large can the derivative of the Riemann zeta function be at its zeros?

More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|$? Surely this is a known result? I can't find anything in the literature about it, though.

Moreover, even if there are extreme values of $\zeta'(\rho)$, is there a nice 'average' for $\sum_{0<\Im{\rho}<T}\zeta'(\rho)^k$? for integer $k$? I ask this as it would help with my understanding of the growth of the derivative.

Answer 1

Bounds for the value of zeta take the form $$ |\zeta(\sigma +it) | \ll t^{ f(\sigma)+\epsilon} $$ where $|t|>2$ and $f$ is some function of $\sigma$. We can take $f(1)=0$ and $f(0)=1/2$ and we can always take $f$ to be convex.

Any bound of this form for $\zeta$ implies an identical form for the derivative of $\zeta$: We can express the derivative of $\zeta$ as the integral of $\zeta$ over a circle and taking the radius of this circle to go to zero slowly the loss between the bound for $\zeta$ and the bound for $\zeta'$ goes to $0$.

There can't be a better bound for $\zeta'$ at an arbitrary point than $\zeta$, since we can also express $\zeta$ as an integral of $\zeta'$. There could conceivably be a better bound for $\zeta'(\rho)$ specifically at zeroes $\rho$ of $\zeta$ but this seems unlikely - we don't typically expect the derivative of a holomorphic function to be especially small at its zeroes.

In particular, if you have no information on the real part of the zero, then the worst case is when $\sigma$ is close to $0$ so $f$ is close to $1/2$ and the best bound will be of the form $t^{1/2+\epsilon}$. Zero-free regions are not currently sufficient to rule out this probably.

Answer 2

As the other answer indicates, at a single zero you can't say much, but there are a lot of results on discrete moments. With $$ J_k(T)=\frac{1}{N(T)}\sum_{0<\gamma<T}|\zeta^\prime(\rho)|^{2k}, $$ Gonek showed that $$ J_1(T)\sim\frac{1}{12}\log(T)^3. $$ Milinovich and Ng showed $$ J_k(T)\gg_k \log(T)^{k(k+2)} $$ and Kirila showed that on RH, $$ J_k(T)\ll_k\log(T)^{k(k+2)}. $$ A search on "moments derivative Riemann zeta" turns up lots of papers.