Riemann set aside proving his famous hypothesis after a few quick attempts
In his 1859 paper on counting primes, Bernhard Riemann remarked that a certain property of his zeta function was very probably true, admitted a rigorous proof would be welcome, and said he had shelved the search after some brief, unsuccessful tries. That offhand aside is now the most celebrated open problem in pure mathematics.
The zeta function takes complex numbers as input and returns complex numbers. It equals zero at every negative even integer, the so-called trivial zeros. It also vanishes at other points, the non-trivial zeros, and those are what matter. The hypothesis says every one of them lies on a single vertical line in the complex plane, called the critical line. Enormous numerical checks agree, but no proof exists.
The function's roots go back to Leonhard Euler, who studied the underlying infinite series in the 1730s while solving the Basel problem. He showed it could be rewritten as a product running over every prime number, which is the deep link between zeta and the primes. Because the series only converges for part of the complex plane, the function must be extended, by a process called analytic continuation, before the hypothesis even makes sense. One oddity of the extended function: its value at zero is minus one half.
Riemann's motive was an exact formula for how many primes lie below a given size, and the zeros appear in that formula. Settling where they sit would therefore sharpen what is known about how primes are distributed. After his death, a note among his papers mentioned an expression for the function that he had never simplified enough to publish. A later mathematician who spent about thirty years proving all but one of the properties Riemann had simply stated admitted nobody had any idea what that expression was. The exception was the hypothesis itself.
David Hilbert grouped it with Goldbach's conjecture and the twin prime conjecture as the eighth of his 23 problems, and the Clay Mathematics Institute offers US$1 million for a solution. Related versions have fallen: André Weil proved the analogue for curves over finite fields.
Source: Riemann hypothesis
