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The Riemann hypothesis (also called the '''Riemann zeta-hypothesis'''), first formulated by Bernhard Riemann in 1859, is one of the most famous and important unsolved problems in Mathematics . It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs. The Riemann hypothesis (RH) is a Conjecture about the distribution of the Zero s of the Riemann Zeta-function ζ(''s''). The Riemann zeta-function is defined for all Complex Number s ''s'' ≠ 1. It has zeros at the negative even integers (i.e. at ''s'' = −2, ''s'' = −4, ''s'' = −6, ...). These are called the trivial zeros. 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 ½. Thus the non-trivial zeros should lie on the so-called critical line ½ + ''it'' with ''t'' a Real Number and ''i'' the Imaginary Unit . The Riemann zeta-function along the critical line is sometimes studied in terms of the Z-function , whose real zeros correspond to the zeros of the zeta-function on the critical line. The Riemann hypothesis is one of the most important open problems of contemporary mathematics, mainly because a large number of deep and important other results have been proven under the condition that it holds. Most mathematicians believe the Riemann hypothesis to be true. ( J. E. Littlewood and Atle Selberg have been reported as skeptical. Selberg's skepticism, if any, waned from his young days. In a 1989 paper, he suggested that an analogue should hold for a much wider class of functions, the Selberg Class ). A $1,000,000 prize has been offered by the Clay Mathematics Institute for the first correct proof. HISTORY 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-trivial zeros of the zeta-function were symmetrically distributed about the line ''s'' = ½ + ''it'', and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(''s'') ≤ 1. In 1896, Hadamard and De La Vallée-Poussin independently proved that no zeros could lie on the line Re(''s'') = 1. Together with the other properties of non-trivial zeros proved by Riemann, this showed that all non-trivial zeros must lie in the interior of the critical strip 0 < Re(''s'') < 1. This was a key step in the first complete proofs of the Prime Number Theorem . In 1900, Millennium prize problems. In 1914, Hardy proved that an infinite number of zeros lie on the critical line Re(''s'') = ½. 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 Number s. Helge Von Koch proved in 1901 that the Riemann hypothesis is equivalent to the following considerable strengthening of the Prime Number Theorem : for every ε > 0, we have | ||
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