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on The Riemann hypothesis

The statement

Let $\zeta(s)$ denote the analytic continuation of $\sum_{n\ge 1} n^{-s}$ to $\mathbb{C}\setminus\{1\}$ . The function has trivial zeros at the negative even integers. The hypothesis asserts that every other zero $\rho$ satisfies

\operatorname{Re}(\rho) = \tfrac{1}{2}.

Equivalently, for every $\varepsilon > 0$ ,

\pi(x) = \operatorname{li}(x) + O\!\left(x^{1/2+\varepsilon}\right),

where $\pi(x)$ counts primes up to $x$ and $\operatorname{li}$ is the logarithmic integral.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.