A Direct Proof of the Prime Number Theorem
... to the right of any singularities in f¯(s).
assume f¯(s) → 0 as |s| → ∞.
Choose ḡ(s) such that f¯(s) − ḡ(s) is analytic at the rightmost
singularity in f¯(s). If ḡ(s) → 0 as |s| → ∞, then shift the integration
contour to the left, apply the Cauchy integral theorem, and get that
f (x) = g(x) + O ...