Proof of the Riemann Hypothesis ( in the final revision process

The Riemann zeta function is defined as ζ(s) = ∞ n=1 1 n s for ℜ(s) > 1 and may be extended to an analytic function on the whole complex plane, except at its unique pole s = 1. The Riemann hypothesis is a conjecture made by Riemann in 1859 asserting that all non-trivial zeros for ζ(s) lie on the line ℜ(s) = 1 2 , which is equivalent to the prime number theorem in the form of π(x)−Li(x) = O(x 1/2 log x), where π(x) = p≤x 1 with the sum runs through the set of primes is the prime counting function and Li(x) = x 2 1 log v d v is Gauss' logarithmic integral function. In this article, we give a proof for the Riemann hypothesis. article may only be reproduced or duplicated for personal or educational use except the publication on Annals of Mathematics is granted.