Riemann hypothesis: a standalone construct

Bernhard Riemann introduced the hypothesis in his 1859 memoir Über die Anzahl der Primzahlen unter einer gegebenen Größe (On the number of primes less than a given magnitude), where he studied the zeta function, the function that sums one over n raised to the power s across the whole numbers and can be extended to almost every complex value of s. He noticed that this extended function has zeros, inputs where its value is zero, and conjectured that every zero other than the trivial ones at the negative even integers lies on a single vertical line in the complex plane, the line where the real part of s equals one half, now called the critical line. The modern authoritative treatment gathered by Borwein, Choi, Rooney, and Weirathmueller (2008) presents the conjecture together with the equivalent statements and partial results that have accumulated since. What makes the hypothesis a landmark is its tight link to the prime numbers: the exact placement of these zeros governs how evenly the primes are distributed among the integers, so a proof would sharpen a large body of number-theory results that currently assume it. It remains unproven, stands as one of the seven Clay Mathematics Institute Millennium Prize Problems named in 2000, and has been checked numerically for many billions of zeros, all of which so far lie on the critical line.[1][2][3]

Unit

A conjecture in analytic number theory about where the zeros of the zeta function lie in the complex plane

Recovery means

Not applicable

Measured by

Assessed by numerical computation of zeta zeros, all found on the critical line so far, and by proofs of equivalent statements; not yet proved in general