# Riemann hypothesis and diophantine equation

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.

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This thread of MathOverflow should help, – Raymond Manzoni Aug 19 '12 at 8:03

## 1 Answer

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.

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What is $\delta(n)$? – Seirios Aug 19 '12 at 8:18
That's a diophantine equation? – Gerry Myerson Aug 19 '12 at 8:52
If $\delta(n)$ is defined as in mathoverflow.net/questions/56376/recovering-n-from-sigman-n then this fails for $n=199$. – John Bentin Dec 13 '12 at 20:49
It looks more like a diophantine inequality. – zerosofthezeta Aug 9 '13 at 5:04