Proof of the Density Hypothesis

The Riemann zeta function is a meromorphic functionon the whole complex plane. It has infinitely many zeros and aunique pole at s = 1. Those zeros at s = −2,−4,−6, . . . areknown as trivial zeros. The Riemann hypothesis, conjectured byBernhard Riemann in 1859, claims that all non-trivial zeros of ζ(s)lie on the line R(s) =12 . The density hypothesis is a conjecturedestimate N(λ, T ) = O(T 2(1−λ)+ǫ) for any ǫ > 0, where N(λ, T ) isthe number of zeros of ζ(s) when R(s) ≥ λ and 0 < I(s) ≤ T ,with12 ≤ λ ≤ 1 and T > 0. The Riemann-von Mangoldt Theoremconfirms this estimate when λ =12 , with T ǫ being replaced bylog T . The xi-function ξ(s) is an entire function involving ζ(s) andthe Euler Gamma function Γ(s). This function is symmetric withrespect to the line R(s) =12 , although neither ζ(s) nor Γ(s) exhibitsthis property. In an attempt to transform Backlund’s proof of theRiemann-von Mangoldt Theorem, from 1918, to a proof of thedensity hypothesis by convexity, we discovered a slightly differentapproach utilizing a modified version of the Euler Gamma function.This modified version is symmetric with respect to the line R(s) =12 , and its reciprocal is an entire function. Aided by this function,we are able to establish a proof of the density hypothesis. Actually,our result is even stronger when 12 < λ < 1, with N(λ, T ) =O(