This paper deals with the characteristics of prime numbers and is divided into four main parts. The first part offers a brief history of the numerous achievements concerning prime numbers. It starts with the very early discoveries of the ancient Greeks and ends with the RSA algorithm. This is followed by a second more mathematical part which states the different definitions of prime numbers and proves their most important characteristics, such as the fundamental theorem of arithmetic.
The last two sections of this paper state the whole way from the proof of infinitely many prime numbers and the first unsure consideration of their distribution by Euclid, Dressler, Euler, Gauss, and Legendre to the still unproven but vitally important Riemann hypothesis. This paper explains how Gauss and Legendre discover the Prime Number Theorem and how Chebyshev nearly managed to prove it. Moreover, it illustrates Euler's discovery of the connection between the zeta function and prime numbers. At the end of the paper, Riemann's Hypothesis about the location of the non trivial zeros of the zeta function for complex numbers and its immense influence not only on the field of prime numbers but also on other areas of mathematics and even physic is elaborated.