Open-problem statements, with their sources attached.

(A) Existence and smoothness of Navier–Stokes solutions on ℝ³.

The four dimensional case of the smooth version of the conjecture is still open. See [Wang2017].

P ≠ NP*:

The conjecture that the complexity classes P and NP are not equal.

(B) Existence and smoothness of Navier–Stokes solutions in ℝ³/ℤ³.

It is conjectured that the only values of $n > 4$ for which the smooth version of the conjecture holds are $n = 5, 6, 12, 56, 61$. See Conjecture 1.17 in [Wang2017].

NP ≠ coNP*:

The conjecture that the complexity classes NP and coNP are not equal.

(C) Breakdown of Navier–Stokes solutions on ℝ³.

(D) Breakdown of Navier–Stokes Solutions on ℝ³/ℤ³.

The Riemann Hypothesis: all non-trivial zeros of the Riemann zeta function have real part $\frac{1}{2}$. That is, if $\zeta(s) = 0$, $s \neq 1$, and $s$ is not a trivial zero $-2(n+1)$ for some $n \in \mathbb{N}$, then $\operatorname{Re}(s) = \frac{1}{2}$.

This is the official Millennium Prize Problem as posed by the Clay Mathematics Institute.

This uses the RiemannHypothesis type from Mathlib, which is defined as ∀ (s : ℂ), riemannZeta s = 0 → (¬∃ n : ℕ, s = -2 * (n + 1)) → s ≠ 1 → s.re = 1 / 2.

The Generalized Riemann Hypothesis asserts that all the non-trivial zeros of the Dirichlet $L$-function $L(\chi, s)$ of a primitive Dirichlet character $\chi$ have real part $\frac{1}{2}$.