The Zeta function according to Riemann

Abstract

In this paper, we present the proof outlined by Riemann in his ground breaking paper from 1859. In his paper, Riemann proves several results about the so called Riemann zeta function and he gives an analytic representation of the prime-counting function pi(x). He closes his ten page paper with a strong approximation of pi(x), which was stronger than the best analytic approximation at that time. Riemann left a lot of the formal proofs to the reader, giving them only hints at the solution, but essentially his proof is incomplete. The formal completion of the proof comes form Hadamard and von Mangoldt. In this paper, we will thoroughly work out Riemann's original approach and we will be able to proof all his assertions except for one switching of summation and integration. The first proof of the legality of this action was given by von Mangoldt, but his proof is indirect and uses a totally different approach than suggested by Riemann. The only knowledge required in this paper, is some background in complex analysis.