The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of Riemann's zeta function ζ(s). It is one of the most important open problems of contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. In June 2004, Louis De Branges de Bourcia claimed to have proved the Riemann hypothesis but this has not yet been confirmed (see below). Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical.)
The Riemann zeta function ζ(s) is defined for all complex numbers s≠1. It has certain so-called "trivial" zeros for s = -2, s = -4, s = -6, ... The Riemann hypothesis is concerned with the non-trivial zeros, and states that:
- The real part of any non-trivial zero of the Riemann zeta function is 1/2.
Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper On the Number of Primes Less Than a Given Magnitude, but as it was not essential to his central purpose in that paper, he did not attempt a proof. Riemann knew that the non-trival zeros of the zeta function were symmetrically distributed about the line z=1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0<=Re(z)<=1.
In 1896 Hadamard and de la Vallée-Poussin independently proved that no zeros could lie on the line Re(z)=1, so all non-trivial zeros must lie in the interior of the critical strip 0<Re(z)<1. This was a key step in the first complete proofs of the prime number theorem.
In 1900 Hilbert included the Riemann hypothesis in his famous list of 23 unsolved problems - it is part of Problem 8 in Hilbert's list. He said of the problem: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?".
In 1914 Hardy proved that an infinite number of zeros lie on the critical line Re(z)=1/2. However, it was still possible that an infinite number (and possibly the majority) of non-trivial zeros could lie elsewhere in the critical strip. Later work by Hardy and Littlewood in 1921 and by Selberg in 1942 gave estimates for the average density of zeros on the critical line.
Recent work has focused on the explicit calculation of the locations of large numbers of zeros (in the hope of finding a counterexample) and placing upper bounds on the proportion of zeros that can lie away from the critical line (in the hope of reducing this to zero).
The Riemann hypothesis and primes
The traditional formulation of the Riemann hypothesis obscures somewhat the true importance of the conjecture. The zeta function has a deep connection to the distribution of prime numbers and Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the following considerable strengthening of the prime number theorem:
The zeros of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicit formulae which show that in the language of Fourier analysis the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes.
The Riemann hypothesis can be generalized in various ways by replacing the Riemann zeta function by the formally similar global L-functions. None of these generalizations has been proven or disproven. See generalized Riemann hypothesis.
Practical Uses of the Riemann hypothesis
The practical uses of the Riemann hypothesis include many equations that have been 'solved' in abstract mathematics with the assumption of the Riemann hypothesis.
A possible proof of the Riemann hypothesis
In June 2004, Louis De Branges de Bourcia of Purdue University, the same mathematician who solved the Bieberbach conjecture, claimed to have proved the Riemann hypothesis in an "Apology for the proof of the Riemann Hypothesis"(pdf). His proof will soon be subjected to review by other mathematicians. De Branges de Bourcia has announced a proof a number of times, but all of his previous attempts at this proof have failed.
The full purported proof is "Riemann Zeta functions" (pdf).
The proof's method has been tried before unsuccessfully, linked is Conrey and Li's counterexample on the problems in the earlier iteration of his proof. 
Possible connection with operator theory
See main article Hilbert-Pólya conjecture
It has long been speculated that the correct way to derive the Riemann hypothesis has been to find a self-adjoint operator, from the existence of which the statement on the real parts of the zeroes of ζ(s) would follow when one applies the criterion on real eigenvalues. This has led to many investigations; but has not yet proven fruitful.
- The Riemann hypothesis
- Computation of zeros of the Zeta function
- The million dollar prize for its solution
- Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire. National Academies Press, 2003. 448 page book at a non-specialist level, can be read online for free.
- Press release regarding the proposed proof
- ZetaGrid A distributed computing project that has verified the Riemann Hypothesis for the first 793 billion nontrivial zeros.
- Marcus Du Sautoy: The Music of the Primes