Millennium Prize Problem · open since 1859
The Riemann Hypothesis
Every non-trivial zero of the zeta function lies on the line Re(s) = ½.
The left side adds up every whole number. The right side multiplies over only the primes. Euler noticed they are equal, which means this one function knows where every prime is. Riemann asked a single question about it, and nobody has answered it in 167 years.
Watch it pass through zero
Feed ζ a complex number s = σ + it and it returns another complex number. Hold σ fixed, let t climb, and the answer traces a path. When σ = ½, the path runs straight through zero again and again: at t ≈ 14.13, 21.02, 25.01 and on forever. Move σ even slightly and the path still loops close, but it never lands.
|ζ(σ + it)| as t climbs · gold dots are the zeros
Every value on this page is computed live in your browser from the formula above. No table of zeros is stored. The zeros you see are found the way mathematicians find them, by watching for the moment the curve changes sign.
The critical strip
ζ also has easy zeros at −2, −4, −6 and so on. These are called the trivial zeros, and they are not the mystery. Every other zero is known to lie in the strip 0 < Re(s) < 1. The zeros are also symmetric about the middle of that strip, because of Riemann’s functional equation:
So a zero at σ + it means there is another at (1 − σ) + it. The hypothesis says the pairs never separate. Every non-trivial zero sits exactly on the centre line, σ = ½.
The primes, rebuilt from the zeros
Here is why it matters. Riemann found an exact formula for the primes in terms of the zeros. Start with a smooth line, then add one wave for each zero. The waves interfere, and the corners of the prime staircase appear.
Each zero ρ = ½ + iγ adds a wave of height √x and frequency γ. If every zero has real part ½, the waves stay small and the primes are as evenly spread as they can be. In precise terms, the gap between the true count of primes and Gauss’s estimate Li(x) is then never much bigger than √x log x. One zero off the line, further right, would make that gap bigger.
How far we have got
- 1737Euler shows that a sum over all whole numbers equals a product over the primes.
- 1859Riemann’s only paper on number theory. He extends ζ(s) to the complex plane, links its zeros to the primes, and writes that it is very likely all the important zeros lie on one line.
- 1896Hadamard and de la Vallée Poussin prove the prime number theorem, using the fact that ζ has no zeros on Re(s) = 1.
- 1900Hilbert puts the hypothesis eighth on his list of problems for the new century.
- 1914Hardy proves that infinitely many zeros lie on the line. Not all, but infinitely many.
- 1972Over tea at Princeton, Hugh Montgomery shows Freeman Dyson how the zeros are spaced. Dyson recognises the pattern at once: it is the spacing of energy levels in heavy atomic nuclei.
- 1989Conrey proves that more than two-fifths of the zeros are on the line.
- 2000The Clay Mathematics Institute offers a million dollars for a proof.
- 2021Platt and Trudgian rigorously confirm every zero up to height 3 × 10¹² sits on the line.
- 2024Guth and Maynard improve an 80-year-old bound on how many zeros could possibly sit off the line.
Trillions of zeros checked, all of them on the line. But checking is not proving. A single exception, however high up, would be enough to make the hypothesis false.
The statement a machine would accept
Mathlib, the Lean library of computer-checked mathematics, already states the hypothesis. A proof would have to make this compile:
def RiemannHypothesis : Prop :=
∀ (s : ℂ) (_ : riemannZeta s = 0) (_ : ¬∃ n : ℕ, s = -2 * (n + 1)) (_ : s ≠ 1), s.re = 1 / 2Statement after the Clay Mathematics Institute. Zeros and ζ values are computed in the browser with Borwein’s algorithm and checked against Odlyzko’s tables. The Lean code is copied from Mathlib/NumberTheory/LSeries/RiemannZeta.lean. See also the other six Millennium Problems.