Is the Riemann hypothesis true?

The question

Prime numbers — 2, 3, 5, 7, 11, 13 and so on, numbers divisible only by one and themselves — look scattered almost at random. Yet on average they obey a law: around a number x, roughly one number in every ln x is prime. The question is how closely the primes follow this law, and how far they can stray from it.

In 1859 Bernhard Riemann wrote an eight-page paper, his only one on number theory. He showed that the exact distribution of the primes is encoded in the zeros of a function now called the Riemann zeta function, and he remarked, almost in passing, that it seemed "very probable" all those zeros lie on one straight line. He couldn't prove it and set the question aside. Nobody has proved it since.

What we know

The zeta function starts as the sum 1 + 1/2^s + 1/3^s + …, extended to complex numbers s. Its "trivial" zeros are at −2, −4, −6 and so on. All other zeros lie in a narrow strip, and the hypothesis says they all have real part exactly 1/2: they lie on the "critical line".
In 1896 Jacques Hadamard and Charles de la Vallée Poussin proved that no zeros lie on the edge of the strip. That was enough to prove the prime number theorem, the average law of the primes.
In 1914 G. H. Hardy proved that infinitely many zeros lie on the critical line. Later work showed that at least 40% of them do. But "most" is not "all", and a single zero off the line would disprove the hypothesis.
Computers have checked the first trillions of zeros: every one lies on the line.
If the hypothesis is true, the primes deviate from their average law as little as possible — roughly by the square root of the numbers involved. Hundreds of published theorems are proved assuming it is true.
A version of the hypothesis for curves over finite fields, algebraic analogues of the ordinary numbers, has been proved: André Weil for curves in the 1940s, Pierre Deligne in full generality in 1974.
In 1900 David Hilbert included the hypothesis among his famous problems, and in 2000 the Clay Mathematics Institute made it one of the seven Millennium Problems, with a prize of one million dollars.

Main hypotheses

The zeros are eigenvalues

Around 1912 Hilbert and Pólya suggested that the zeros might be the energy levels of some physical system. Such levels are real numbers by nature, which would force the zeros onto the line. In 1972 Hugh Montgomery found that the gaps between zeros follow the same statistics as the energy levels of heavy atomic nuclei, described by random matrices. Andrew Odlyzko later confirmed it numerically with great precision. The system itself hasn't been found.

Transferring Deligne's proof

The finite-field version was proved with powerful geometry. Many mathematicians hope to build a similar geometry for ordinary numbers, but nobody knows what "the curve" behind ordinary integers should be.

It might be false

A minority doubts it. Zeros high up the line come very close to each other, and a counterexample could be too large for any computer to reach. For some related functions with similar properties, analogous statements turn out to be false.

Where understanding ends

We have enormous evidence and no proof. The trillions of checked zeros mean less than they seem: in number theory, patterns sometimes hold for astronomically large numbers and then break, so numerical evidence can't settle the question.

The deeper gap is that we don't understand why the zeros would lie on the line. The known methods give either "most zeros" or "no zeros in large regions", and there is no sign that improving them would reach "all". In 2024 Larry Guth and James Maynard proved the first significant improvement in decades on how many zeros can lie away from the line. It was celebrated precisely because progress is so rare, and it still falls far short of the full statement.

Mathematicians generally believe that a proof will require a new idea about the structure of whole numbers, not a cleverer calculation.

What would settle it

A proof: most likely through a new mathematical structure that explains the connection between the zeros, random matrices and the geometry that worked for finite fields.
A counterexample: a single zero off the critical line, found by calculation or constructed theoretically.
A proof that the hypothesis can't be proved or disproved from the standard axioms. This is considered unlikely for this kind of statement: if it were unprovable, there could be no counterexample, and the hypothesis would effectively be true.