The Riemann Hypothesis

The statement. The Riemann zeta function is ζ(s) = Σ 1/nˢ for Re(s) > 1, extended to the whole complex plane. Apart from the “trivial” zeros at −2, −4, −6, …, the hypothesis says every zero has real part exactly ½.

What it means in plain words. The primes look random but are not. Their count up to x is close to x/ln x, and the Riemann Hypothesis says the error in that estimate is as small as it could possibly be — the primes are spread as evenly as anything random could be. Ten thousand million zeros have been checked; none has broken the rule; no one has proved it cannot happen.

What you need first. Stage 10 (primes, the prime counting idea), Stage 6 (integrals and series), Stage 12 (convergence of Σ 1/nˢ, and complex functions — Euler's formula, analytic continuation). That is the whole path from arithmetic to being able to read the statement.

The parts. (1) Why Σ 1/nˢ and the primes are the same object — Euler's product ζ(s) = Π 1/(1 − p⁻ˢ), which is the fundamental theorem of arithmetic written as an equation. (2) Why the series, which only converges for Re(s) > 1, still defines a function everywhere else (analytic continuation) and satisfies a symmetry about the line Re(s) = ½ (the functional equation). (3) Why the zeros control the primes — the explicit formula writes the prime count as a sum over zeros. (4) The hypothesis itself: all those zeros on one line.

What is known. Infinitely many zeros lie on the line (Hardy, 1914); at least 40% of them do (Conrey); the first 10¹³ do (computation); the analogous statement for finite fields is a theorem (Deligne, 1974). Nobody knows how to attack the general case, which is why it is a Millennium Prize problem.