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## Riemann hypothesis
The
The Riemann zeta function ζ(
Thus the non-trivial zeros should lie on the so-called 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:
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 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 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. ## External links- The Riemann hypothesis
- 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.
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