This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.
Bernhard Riemann did not set out to write mathematics' most famous unsolved problem. His 1859 paper, barely eight pages long, was really about counting primes: he wanted a precise formula for how many primes exist below a given number, and in the course of getting there he needed to understand a function built from an infinite sum, now called the Riemann zeta function, and specifically where that function equals zero. Almost as an aside, he conjectured that every one of its "interesting" zeros, the nontrivial ones, sits on a single vertical line in the complex plane, real part exactly one half. He did not prove it. Nobody has since. What makes the Riemann Hypothesis unusual, even among famous open problems, is how much other mathematics already assumes it is true. Hundreds of published theorems begin "assuming the Riemann Hypothesis" and derive a further result conditionally; if the hypothesis is ever proved, all of that conditional mathematics becomes unconditional overnight, and if it is ever disproved, an unknown amount of it collapses. Computational searches have checked the first many trillions of zeros without finding a single exception, which sounds like overwhelming evidence, and in an ordinary empirical science it might be. But mathematics does not accept a pattern, however many trillion times confirmed, as a substitute for a proof covering every case, because the history of number theory is dotted with patterns that held for enormous stretches and then broke: certain prime-counting inequalities were once conjectured to hold forever and were later shown, by an argument that produces no actual counterexample small enough to check by hand, to fail somewhere past numbers too large to write down. The Clay Mathematics Institute named it one of seven Millennium Prize Problems in 2000, each carrying a one million dollar award, not because a cash prize was needed to attract attention (mathematicians had already spent a century and a half on it) but because it named, plainly, what the field considers unfinished business of the highest order. A century and a half after Riemann's aside, it still is.