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The Riemann Hypothesis

millennium prize problems / the riemann hypothesis problem

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function z(s) called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation

z(s) = 0

lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

Mathematical Description authored by Enrico Bombieri