Riemann hypothesis and diophantine equation

Question

I read that showing Riemann hypothesis is true was equivalent to showing a particular diophantine equation doesn't have any solution.

Is there an explicit example of such a diophantine equation?

EDIT: In slides of Poonen, it is shown how DPRM theorem implies that there exists an polynomial equation that has a solution in integers if and only if the Riemann hypothesis is false. However, no explicit equation is given.

Answer 1

Yes. The Riemann Hypothesis ends up being equivalent to the following:

$$\forall n > 0, \left( \sum_{k \leq \delta(n)} \frac{1}{k} - \frac{n^2}{2} \right)^2 < 36n^3$$

This is proven using the same techniques from the proof of Hilbert's 10th Problem, and this is proven in the book Mathematical Developments Arising From Hilbert Problems with respect to the 10th problem.