1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking

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) = 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, 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.

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 1.

If I were to awaken after having slept for five hundred years, my first question would be: Has the Riemann hypothesis been proven? [1]