Riemann zeta function

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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 a prominent function of great significance in number theory. It is a named after German mathematician Bernhard Riemann. It is so important because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics.

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]

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