A Proof for the Riemann Hypothesis

The Riemann zeta function is defined as ζ(s) = ∑∞ n=1 1 ns for ℜ(s)> 1 and extended to an analytic function on the whole complex plane excluding its unique pole at s = 1. The Riemann hypothesis ass?? 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 +ǫ) for any positive ǫ, where π(x) = ∑ p≤x 1 with the sum runs through the set of primes is the prime counting function and Li(x) = ∫ x 1 2 log v dv is Gauss ’ logarithmic integral function. In this article, we prove a stronger result related to the prime number theorem so that validify the Riemann hypothesis.