A Proof for the Density Hypothesis

The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for R(s) > 1 and may be extended to a regular function on the whole complex plane excluding 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 has a broad application in every branch of mathematics. The density hypothesis is a related “weaker” conjecture about the estimate of the number of zeros for the Riemann zeta function in the so-called critical strip 0 ≤ R(s) ≤ 1. In this article, we give a proof for the density hypothesis 1. 2000 Mathematics Subject Classification: 11M26, 30C80.