The Riemann Hypothesis
We reach the summit — the most famous unsolved problem in mathematics. The Riemann
Hypothesis is a single, crisp claim about where the
zeta function
equals zero. It has stood unproven since 1859, carries a million-dollar
prize, and would, if true, tell us almost everything about the fine distribution of the primes.
The two kinds of zeros
The analytically continued \zeta(s) has zeros of two kinds. The
trivial zeros sit at the negative even integers
s = -2, -4, -6, \dots and are well understood. The
non-trivial zeros all lie in the critical strip
0 < \Re(s) < 1 — and these are the ones that govern the primes.
The hypothesis
Every non-trivial zero of \zeta(s) lies on the critical line
\Re(s) = \tfrac{1}{2}.
Over 10^{13} zeros have been computed, and every single one lies exactly
on this line. Yet a proof that they all do — that none ever strays — has eluded every
mathematician for more than 160 years. It is one of the seven Clay
Millennium Prize Problems.
Why it matters so much
The location of the zeros controls the error term in the
Prime Number Theorem.
The Riemann Hypothesis is exactly the statement that the primes are as
regularly distributed as they possibly could be — that
\pi(x) hugs \operatorname{Li}(x) with the
smallest conceivable error:
|\pi(x) - \operatorname{Li}(x)| = O\!\left(\sqrt{x}\,\ln x\right).
Hundreds of theorems across mathematics begin "assume the Riemann Hypothesis…". It is the keystone
the whole arch of analytic number theory leans on — known to hold for every zero we can reach, and
waiting, still, for a proof. A fitting place to end: number theory's simplest objects, the whole
numbers, guarding its deepest secret.