  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.
