In 1740 Leonhard Euler introduced the zeta function as an infinite series.

In 1859 the German mathematician Bernhard Riemann treated the Zeta Function as a function of a complex variable z.

Where z = a + bi

a = Re(z) the real part

bi = Im(z) the imaginary part

For this reason this form of Zeta Function is known as a Riemann Zeta Function.

Related to the Riemann Zeta Function is the Riemann hypothesis which to this date still remains unproved.

The function has no zeros in Re(z) ≥ 1; and for Re(z) ≤ 0 it has zeros at z = -2, -4, -6, ...; but for 0 < Re(z) < 1 there is an infinite number of zeroes, which are called nontrivial zeroes.

The hypothesis is that all the nontrivial zeros of