Riemann hypothesis

The Riemann zeta function, usually written ζ(s), is initially defined by the series 1 + 1/2^s + 1/3^s + … when the real part of the complex number s is greater than 1, and it can be extended to most other complex values by analytic continuation. It has zeros at the negative even integers, called trivial zeros. The Riemann hypothesis concerns its other, nontrivial zeros: it asserts that each lies on the critical line, where the real part of s equals 1/2. These zeros lie in the critical strip, with real part between 0 and 1.

The zeta function encodes information about prime numbers through its relationship to an infinite product over primes. The prime number theorem gives the leading, average pattern for how primes occur among positive integers. The Riemann hypothesis would constrain the error in approximations to prime-counting, giving a precise form of control over departures from that average. It would have consequences across number theory, but it is not a formula for predicting individual primes.

Bernhard Riemann introduced the conjecture in an 1859 paper on the number of primes below a given size. It remains one of the Clay Mathematics Institute's seven Millennium Prize Problems. Computations have verified very large finite ranges of zeros, but checking a finite set cannot establish a claim about every nontrivial zero. As of 2026, the conjecture remains unproved and undisproved.