The Riemann Hypothesis
$1,000,000 USD
1859
Bernhard Riemann
Open
TL;DR
The Riemann Hypothesis is a conjecture about the distribution of prime numbers. Primes look random up close — 2, 3, 5, 7, 11, 13, 17… — but zoom out and they follow a stunning pattern, described by the prime number theorem. The Riemann zeta function ζ(s) is the mathematical object that captures this pattern.
Riemann noticed that the "non-trivial zeros" of ζ(s) — the complex numbers where ζ(s) = 0, other than at negative even integers — all seemed to have real part exactly ½. He conjectured this holds for every such zero, along a vertical line in the complex plane called the critical line.
If true, this pins down the primes with extraordinary precision. Hundreds of theorems in number theory begin "Assuming the Riemann Hypothesis…" — they'd become unconditional overnight if it were proved. Trillions of zeros have been computed numerically and every single one lies on the critical line. But numerical evidence is not proof, and the hypothesis has stood open for more than 165 years.
Formal statement
Every non-trivial zero of the Riemann zeta function has real part equal to .
Why it matters
The distribution of primes underlies cryptography, random number generation, and vast swathes of pure mathematics. RH is the deepest statement we have about that distribution — and its resolution would either confirm an entire framework of number-theoretic reasoning, or shatter it.
History
Bernhard Riemann introduced the hypothesis in a single 1859 paper titled "On the Number of Primes Less Than a Given Magnitude" — his only paper on number theory. In 8 pages he sketched the connection between ζ(s) and prime counting, made his conjecture almost in passing, and moved on.
Hilbert placed the Riemann Hypothesis at #8 on his famous list of 23 problems in 1900. He is quoted as saying that if he awoke after sleeping for 500 years, his first question would be whether the Riemann Hypothesis had been proved.
Godfrey Hardy proved in 1914 that infinitely many zeros lie on the critical line — but that leaves open the possibility that some don't. Atle Selberg strengthened this in 1942 by showing a positive proportion of zeros are on the line. In 1974, Norman Levinson pushed the proportion to over one-third. Brian Conrey improved it to 40% in 1989. That's roughly where we are today: we know a large chunk of zeros are on the critical line, but not all.
Numerical verification has exhausted the first 10 trillion zeros. All on the line.
Current status of research
Deep connections have been established between RH and:
- Random matrix theory — the statistical distribution of ζ zeros matches the eigenvalues of large random Hermitian matrices (Montgomery, 1972; Odlyzko's numerical confirmation).
- Explicit formulae — the primes and the zeros are Fourier duals of each other.
- The Selberg trace formula — an analog for hyperbolic surfaces suggests a deeper geometric interpretation might exist.
The Hilbert–Pólya conjecture posits that the zeros correspond to eigenvalues of some self-adjoint operator, which would automatically force them onto the real line (and hence, after a shift, the critical line). Finding such an operator would prove RH — but no candidate has been constructed.
Notable attempts
Louis de Branges has announced proofs multiple times; none have been accepted. Michael Atiyah announced a proof at the 2018 Heidelberg Laureate Forum; it was widely dismissed by specialists within days. The pattern reflects the problem's depth — every generation produces sincere attempts, none succeed.
What a resolution would mean
A proof would unconditionally confirm hundreds of number-theoretic results currently conditional on RH: sharpest possible bounds on prime gaps, estimates for arithmetic progressions of primes, the largest known lower bounds on class numbers of imaginary quadratic fields, and much more.
A disproof — finding even one zero off the critical line — would be equally revolutionary, invalidating a century of conditional results and forcing a complete reconstruction of analytic number theory.