Riemann Hypothesis

The Riemann hypothesis is the single most important unsolved problem in mathematics. First proposed in Bernhard Riemann’s 1859 paper “On the Number of Prime Numbers less than a Given Magnitude,” it concerns the zeros of the Riemann zeta function:

$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$

The zeta function, originally defined as an infinite sum for complex $s$ with real part greater than 1, can be extended to the entire complex plane (except $s = 1$). It has two kinds of zeros:

• Trivial zeros at the negative even integers $-2, -4, -6, \ldots$
• Non-trivial zeros in the “critical strip” $0 < \Re(s) < 1$

Riemann conjectured that every non-trivial zero has real part exactly $\tfrac{1}{2}$ — that is, they all lie on the “critical line.”

Why it matters

The Riemann hypothesis is far more than a curiosity about a particular function. Its truth would give the sharpest possible error estimate in the prime number theorem, which describes how primes are distributed among the integers. Hundreds of other number-theoretic results are proved “conditional on RH”: if RH holds, then such-and-such theorem follows. A proof would cascade.

The state of play

As of 2025:

• The first $10^{13}$ non-trivial zeros have been computed and verified to lie on the critical line.
• G.H. Hardy proved in 1914 that infinitely many zeros lie on the line.
• Selberg and others have shown that a positive proportion of zeros lie on the line.
• Most number theorists believe RH is true — but no one has a proof.

The prize

The Riemann hypothesis is one of the seven Millennium Prize Problems announced by the Clay Mathematics Institute in 2000. A rigorous proof (or disproof) carries a one-million-dollar reward.

It is also Problem 8 on Hilbert’s 1900 list — making it one of the few problems to appear on both the century-opening and century-closing “great unsolved” lists.

Beyond mathematics

Deep connections have been discovered between the zeta function’s zeros and random matrix theory — the eigenvalue spacings of large random Hermitian matrices match the spacings between zeta zeros remarkably well. This has led to conjectures that the zeros encode the spectrum of some unknown quantum system, a speculation pursued under the name quantum chaos.

Whatever the ultimate origin of the hypothesis’s truth (or failure), it has become a Rosetta Stone connecting number theory, analysis, and mathematical physics.

Frequently asked

Why does this conjecture matter so much?

Because hundreds of results in number theory are proved 'conditional on the Riemann hypothesis' — if it is true, a cascade of consequences follows. Its truth would also give us the sharpest possible bound on the error in the prime number theorem.

Has there been any progress?

Yes. The first 10^13 zeros have been computed and verified to lie on the critical line. Hardy proved in 1914 that infinitely many zeros lie on the line. Selberg and others have shown that a positive proportion of zeros lie there. But a proof that all non-trivial zeros lie on the critical line remains elusive.

Continue reading

Formula

The Riemann Zeta Function

An infinite sum that hides the distribution of the primes. The object at the heart of the most important unsolved problem in mathematics.

Read →Person

Bernhard Riemann

German mathematician of extraordinary depth. In a brief life he transformed analysis, number theory, and differential geometry, and posed the hypothesis that remains mathematics' most famous open problem.

Read →Blog

Stirling's Approximation: How Big Is n Factorial, Really?

The factorial n! grows faster than any polynomial but slower than any exponential. A single formula from 1733 pins down its size precisely — and inexplicably involves both π and e.

Read →