Clay Mathematics Institute

Dedicated to increasing and disseminating mathematical knowledge

Riemann Hypothesis

Zeta Function

 

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

    ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ...

called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation

    ζ(s) = 0

lie on a certain vertical straight line.

This has been checked for the first 10,000,000,000 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.

This problem is: 
Unsolved