A proof for the Riemann hypothesis

The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for R(s) > 1 and can be extended to a regular function on the whole complex plane deleting its unique pole at 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) = 12 , which is equivalent to the prime number theorem in the form of π(x)−Li(x) = O(x 1 2 ) 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 2 1 log v d v is Gauss’ logarithmic integral function. In this article, it gives a proof for the density hypothesis and so that settles the long time due justification for the Riemann hypothesis from the equivalence of the density hypothesis and the Riemann hypothesis proved recently in [12], which in turn gives a prime number theorem stated as above. 2000 Mathematics Subject Classification: 11M26, 30C80.