Riemann hypothesis

In mathematics, the Riemann hypothesis (also called the Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems. 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.

Related Topics:
Mathematics - Bernhard Riemann - 1859

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta function ζ(s). The Riemann zeta function 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:

Related Topics:
Conjecture - Zero - Riemann zeta function - Complex number

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: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.

Related Topics:
Real number - Imaginary unit - Z function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The Riemann hypothesis 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. (J. E. Littlewood and Atle Selberg have been reported as skeptical.) In 2004, Xavier Gourdon and Patrick Demichel verified the Riemann hypothesis through the first ten trillion non-trivial zeros using the Odlyzko-Schönhage algorithm.

Related Topics:
Mathematics - Clay Mathematics Institute - J. E. Littlewood - Atle Selberg - Xavier Gourdon - Patrick Demichel - Odlyzko-Schönhage algorithm

~ ~ ~ ~ ~ ~ ~ ~ ~ ~