Proof of the strong Density Hypothesis

The Riemann hypothesis, conjectured by Bernhard Riemann in 1859, claims that the non-trivial zeros of $zeta(s)$ lie on the line $Re(s) =1/2$. The density hypothesis is a conjectured estimate $N(lambda, T) =Obigl(Tsp{2(1-lambda) +epsilon} bigr)$ for any $epsilon >0$, where $N(lambda, T)$ is the number of zeros of $zeta(s)$ when $Re(s) gelambda$ and $0 <Im(s) le T$, with $1/2 le lambda le 1$ and $T >0$. The Riemann-von Mangoldt Theorem confirms this estimate when $lambda =1/2$, with $Tsp{epsilon}$ being replaced by $log T$. In an attempt to transform Backlunds proof of the Riemann-von Mangoldt Theorem to a proof of the density hypothesis by convexity, we discovered a different approach utilizing an auxiliary function. The crucial point is that this function should be devised to be symmetric with respect to $Re(s) =1/2$ and about the size of the Euler Gamma function on the right hand side of the line $Re(s) =1/2$. Moreover, it should be analytic and without any zeros in the concerned region. We indeed found such a function, which we call pseudo-Gamma function. With its help, we are able to establish a proof of the density hypothesis. Actually, we give the result explicitly and our result is even stronger than the original density hypothesis, namely it yields $N(lambda, T) le 8.734 log T$ for any $1/2 < lambda < 1$ and $Tge 2445999554999$.