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Riemann zeta function

Riemann zeta function ζ(s) in the complex plane. The color of a point s shows the value of ζ(s): strong colors are for values close to zero and hue encodes the value's argument. The white spot at s= 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re(s) = 1/2 are its zeros.
The coloring of the complex function-values used above: positive real values are presented in red.

In mathematics, the Riemann zeta function is an important function in number theory. It is related to the distribution of prime numbers. It also has uses in other areas such as physics, probability theory, and applied statistics. It is named after the German mathematician Bernhard Riemann, who wrote about it in the memoir "On the Number of Primes Less Than a Given Quantity", published in 1859.

The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta function. Many mathematicians consider the Riemann hypothesis to be the most important unsolved problem in pure mathematics. [1] In the year 2000, the Clay Mathematics Institute included the Riemann hypothesis as one of their Millennium Prize Problems, promising a reward of US$1 million to anyone who could solve it.

  1. ↑ Riemann, Bernhard. "On the Number of Prime Numbers less than a Given Quantity" (PDF). Archived from the original (PDF) on 2016-03-04. Retrieved 2016-03-09.
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