Riemann hypothesis

Remarks

The zeros of $\zeta$ are the trivial zeros $-2, -4, -6, \dots$ and the nontrivial zeros, which lie in the strip $0 < \operatorname{Re} s < 1$. This follows from the functional equation and the non-vanishing on the line Re s = 1. So the conjecture says that every nontrivial zero lies on the critical line $\operatorname{Re} s = 1/2$. The first nontrivial zeros are at about $\frac{1}{2} \pm 14.1347 i$. The conjecture is open. Riemann stated it in his 1859 paper on the number of primes below a given size. It is one of Hilbert's problems of 1900 and one of the Clay Millennium Prize Problems. Known partial results: Hardy proved that infinitely many zeros lie on the critical line, and Selberg proved that a positive proportion of them do. An equivalent form: $\pi(x) = \operatorname{li}(x) + O(\sqrt{x} \log x)$, where $\pi$ is the prime-counting function and $\operatorname{li}(x) = \int_2^x dt / \log t$. The generalized Riemann hypothesis states the same for Dirichlet L-functions and other L-functions.