Let $\zeta(s) $ be the Riemann zeta function. Let $\zeta’(s) $ and $\zeta’’(s) $ be the first and second derivative of that Riemann zeta function.
In analytic number theory I see the use of $\zeta(s) $ and $\zeta’(s)$ or combinations of them alot.
But I wonder about the potential of $\zeta’’(s) $ ?
I assume it has been investigated ?
Should we consider it more ?
The second derivative comes up as well. For example, the number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function is interesting in general, not only for the first derivative. As a reference, see the paper On the Zeros of the Second Derivative of the Riemann Zeta Function under the Riemann Hypothesis.
Questions on the second derivative at MSE:
Alternative form to express the second derivative of $\zeta (2) $
Derivatives of the Riemann zeta function at $s = 1/2$
Derivative of Riemann zeta, is this inequality true?