First published in Riemann (1859), the Riemann hypothesis states that the nontrivial Riemann zeta function zeros all lie on the "critical line" , where denotes the real part of s. The Riemann hypothesis is also known as Artin's conjecture.
The hypothesis has thus far resisted all attempts to prove it. It has been computationally tested and found to be true for the first zeros (Brent et al. 1982, which covered zeros in the region ). More recently, S. Wedeniwski uses ZetaGrid, an internal computer cluster of IBM Corporation combined with external computations compiled on
http://www.zetagrid.net/ to prove that the first nontrivial zeros of the lie on the critical line. This computation verifies that the Riemann hypothesis is true at least for all .
In 2000, Clay Mathematics Institute (
http://www.claymath.org/) offered a $1 million prize (
http://www.claymath.org/Millennium_Prize_Problems/Rules_etc/) for proof of the Riemann hypothesis. Interestingly, disproof of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the critical line), does not earn the $1 million award.
The Riemann hypothesis is equivalent to the statement that all the zeros of the Dirichlet eta function (a.k.a. the alternating zeta function)