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Riemann hypothesis

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. Most mathematicians believe the Riemann hypothesis to be true.

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.

Thus the non-trivial zeros should lie on the so-called critical line 1/2 + it with t a real number and i the imaginary unit.

This traditional formulation 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:

<math>\pi(x)\rightarrow\int_2^x \frac{dt}{\ln(t)} + O\left(\sqrt x\,\ln(x)\right)\quad{\rm as} \quad x\rightarrow\infty</math>
where, π(x) is the prime-counting function, ln(x) is the natural logarithm of x, and the O-notation is the Landau symbol.

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 have been proven or disproven. See generalized Riemann hypothesis.

Hilbert Polya conjecture

Hilbert and Polya speculated that values of t such that 1/2 + it is a zero of the zeta function might be the eigenvalues of a Hermitian operator, and that this would be a way of proving the Riemann hypothesis. At the time, there was little basis for such speculation. However Selberg in the early 1950s proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian. This so-called Selberg trace formula bore a striking resemblance to the explicit formulae, which gave credence to the speculation of Hilbert and Polya.

Hugh Montgomery investigated and found that the statistical distribution of the zeros on the critical line has a certain property. The zeros tend not to be too closely together, but to repel. Visiting at the Institute for Advanced Study in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices, which is of importance in physics due to the fact that the eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics.

Dyson saw that the statistical distribution found by Montgomery was exactly the same as the pair correlation distribution for the eigenvalues of a random Hermitian matrix. Subsequent work has strongly born out this discovery, and the distribution of the zeros of the Riemann zeta function is now believed to satisfy the same statistics as the eigenvalues of a random Hermitian matrix, the statistics of the so-called Gaussian Unitary Ensemble[?]. Thus the conjecture of Polya and Hilbert now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.

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