The Analytic Expression for Riemann's Prime Counting Function via the Residue Theorem

TL;DR

This paper derives Riemann's prime counting function expression using the Residue Theorem by constructing a suitable branch of log(zeta) and evaluating the resulting integral, confirming Riemann's original formula.

Contribution

It introduces a method to evaluate Riemann's prime counting function via the Residue Theorem with a carefully constructed branch of log(zeta).

Findings

Derived an integral representation of F(x) using the Residue Theorem.

Constructed a holomorphic branch of log(zeta) extending to the left half-plane.

Confirmed the integral evaluation matches Riemann's original expression.

Abstract

In his paper "On the Number of Primes Less Than a Given Magnitude", Bernhard Riemann introduced a prime counting function F(x) which counts the number of primes under x. Riemann obtained an analytic expression for F(x) by evaluating an inverse Laplace Transform. His method involved advanced techniques of analysis. However, this transform can be evaluated using the Residue Theorem when an appropriate branch of log(zeta) is defined. In this paper, a method for constructing a holomorphic branch of log(zeta) extending to the left half-plane is described along with it's geometry surrounding the logarithmic branch points. Using this information, an integral representation of F(x) is formulated in terms of this branch of log(zeta) which is then evaluated. The results are shown equal to Riemann's expression.