What Is the Riemann Hypothesis?

The Riemann zeta function ζ(s) extends the familiar sum 1 + 1/2^s + 1/3^s + … to the entire complex plane. It has infinitely many “trivial” zeros at the negative even integers, and infinitely many “non-trivial” zeros elsewhere. Bernhard Riemann conjectured in 1859 that every one of those non-trivial zeros has real part exactly 1/2 — that they all sit on a single vertical line in the complex plane, the critical line. That is the Riemann hypothesis.

Why anyone cares where a function’s zeros sit

Riemann showed that the zeta function’s zeros are tightly linked to the distribution of prime numbers — how evenly or unevenly primes are spread out as numbers get large. The Prime Number Theorem describes that distribution on average; the Riemann hypothesis would pin down the size of the error term as tightly as is theoretically possible. A huge amount of modern number theory is already built on the assumption that it’s true.

Checked, not proven

Computers have verified that the first many trillion non-trivial zeros all lie exactly on the critical line, with no counterexample ever found. That kind of overwhelming numerical evidence is common for open problems in number theory — it makes the conjecture more credible, but a single zero found off the line anywhere among infinitely many candidates would disprove it, and no amount of checking can rule that out.

Why it’s here

The Riemann hypothesis is one of the seven Millennium Prize Problems, with a $1,000,000 award from the Clay Mathematics Institute attached. It’s tracked in the Registry as its own record, with the exact formal statement, rather than folded only into the roundup of all seven.

Primary source

Clay Mathematics Institute, “Riemann Hypothesis” ↗