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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.

In 2004, Xavier Gourdon verified the Riemann Hypothesis through the first ten trillion non-trivial zeros using the Odlyzko-Schönhage algorithm .

1 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-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 HadamardJacques Solomon Hadamard ( December 8, 1865 October 17, 1963) was a mathematician best known for his proof of the prime number theorem. He studied at the Ecole Normale Superieure. After the Dreyfus affair, which involved him personally, he became politica 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 theoremIn number theory, the prime number theorem PNT describes the approximate, asymptotic distribution of the prime numbers. For any positive real number x define : The prime number theorem then states that : where ln x is the natural logarithm of x''. This no.

In 19001900 is the common year starting on Monday. see link for calendar) For the film, see 1900 (film). Events January January 1 Nigeria becomes British protectorate January 2 John Hay announces the Open Door Policy to promote trade with China. January 2 Chicag HilbertDavid Hilbert ( January 23, 1862 February 14, 1943) was a German mathematician born in Wehlau, near Konigsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th cen included the Riemann hypothesis in his famous list of 23 unsolved problemsHilbert's problems are a list of 23 problems in mathematics put forth by German mathematician David Hilbert in the Paris conference of the International Congress of Mathematicians in 1900. The problems were all unsolved at the time, and several of them tu - 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 1914Events January 4 77 seal hunters freeze to death on ice near Labrador January 5 Ford Motor Company announces an eight-hour workday and a minimum wage of $5 for a day's labor February 13 Copyright: In New York City the ASCAP (for American Society of Compos HardyGodfrey Harold Hardy ( February 7, 1877 December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. Non-mathematicians know him for two things: A Mathematician's Apology his essay from 19 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).

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Riemann hypothesisThe 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 ha Riemann zeta functionIn mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. It also has applications in physics. Definition The Riemann zeta function ζ s is Riemann mapping theoremThe Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C then there exists a bijective holomorphic conformal map f : U D where D { z in C : z pi;, π Riemann integralIf you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. In a branch of mathematics known as real analysis, the Riemann integral created by Bernhard Riemann, was the first rig Riemannian manifoldIn Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. This allows one to define of various notions as the length Riemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. Riemann surfaces can be thought of as a "deformed versions" of the complex plane: locally near every point they look like patches of the complex pla Riemann sum Riemann sphere Riemannian geometry Riemann-Roch theorem Riemann-Stieltjes integral Riemann-Hurwitz formula Riemann-Liouville differintegral Riemannian submanifold