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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 ?

mick
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    The elementary proof of the PNT by Selberg and Erdos can be seen as a study of the behaviour of $\zeta''(s)$ in a neighbourhood of $s=1$. Trivially $\zeta''$ is involved in series like $\sum_{n\geq 1}\frac{\log^2(n)}{n^s}$ or $\sum_{n\geq 1}\frac{n^s}{(e^{\pi s}\pm 1)^2}$. – Jack D'Aurizio Sep 09 '18 at 18:52
  • How is the second Sum related Jack ? – mick Sep 09 '18 at 19:03
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    Via the Mellin transform. See this question, for instance, where a $\zeta'(5)$ suddenly appears. – Jack D'Aurizio Sep 09 '18 at 19:11
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    It may be of interest to note that Levinson and Montgomery have a paper regarding the zeros of not just the second derivative, but of the $k^{th}$ derivative. – Clayton Sep 11 '18 at 17:29

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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?

Dietrich Burde
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  • Nice and useful answer. But most of these ideas are analytic computations and boundaries that hold If RH is true. If’s but not many iffs. So Mainly no equivalents to RH and no things related to primes apart from RH. I assume more focus on primes does exist ? Thank you +1. – mick Sep 09 '18 at 19:42
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    Yes, there are indeed statements on $\zeta''(s)$ being equivalent to RH, see for example here. – Dietrich Burde Sep 10 '18 at 10:55