Riemann Hypothesis

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The Riemann Hypothesis, one of the most important unsolved problems in Mathematics, relates to the non-trival zeros of the Riemann Zeta function, stating that all such zeros have Re(s) = 1/2, lying on the so-called critical line.

This has been tested for the first $\small{10^{13}}$ roots but that gets us no nearer to proving the hypothesis.

The Riemann zeta function is intimately related to the distribution of prime numbers, and it turns out that the Riemann hypothesis is equivalent to the following statement:

where $\pi(x)$ is the number of prime numbers below x and Li is the logarithmic integral function.