Is the Riemann hypothesis true?

In 1859, Bernhard Riemann published an eight-page paper on how prime numbers are distributed, and remarked in passing that a certain property of a certain function was "very probably" true, though he had set the question aside after a few fleeting attempts. That aside is now the most famous unproved claim in mathematics. It has outlived every mathematician who has attacked it, survived a Millennium Prize bounty of one million dollars, and been verified by computer in trillions of individual cases, which in mathematics counts for exactly nothing.

Why it matters

The primes are the atoms of arithmetic, and they arrive irregularly: sometimes clustered, sometimes strangely absent. Riemann’s zeta function converts that irregularity into geometry. The function has a set of special points called zeros, and the hypothesis says they all lie on a single vertical line. If that is true, the primes are as evenly distributed as they could possibly be; their apparent randomness is perfectly disciplined. Hundreds of published theorems begin "assuming the Riemann hypothesis," so a proof would confirm an entire wing of mathematics in one stroke, and a disproof would demolish it. The primes also underpin practical cryptography, though a proof would validate current understanding rather than break any codes.

What has been tried

Hardy proved in 1914 that infinitely many zeros lie on the critical line, and later work pushed that to a positive fraction of them, currently at least two fifths. Computation has verified the first ten trillion or so zeros, every one on the line. The most tantalizing lead came by accident in the 1970s, when Montgomery’s formula for spacings between zeros turned out to match the statistics of energy levels in heavy atomic nuclei, exactly the pattern random matrix theory predicts. That revived an old idea of Hilbert and Pólya: the zeros might be the spectrum of some undiscovered physical operator, making the hypothesis a fact about a hidden kind of physics. No one has found the operator.

Where the edge is

There is no consensus even on which field the proof will come from: analysis, algebraic geometry, or physics. Analogues of the hypothesis in simpler settings were proved by Weil and Deligne, and those proofs are among the deepest known, yet the techniques have not transferred. The gap between "true in every checked case" and "true" remains absolute.

What would count as an answer

A proof or a counterexample. Mathematics grants no partial credit here: one zero off the line settles it one way, one finished argument settles it the other.