Proof of the strong Density Hypothesis

Abstract

The Riemann hypothesis, conjectured by Bernhard Riemann in 1859, claims that the non-trivial zeros of $ζ(s)$ lie on the line $\Re(s) =1/2$. The density hypothesis is a conjectured estimate $N(λ, T) =O\bigl(T\sp{2(1-λ) +ε} \bigr)$ for any $ε>0$, where $N(λ, T)$ is the number of zeros of $ζ(s)$ when $\Re(s) \geλ$ and $0 <\Im(s) \le T$, with $1/2 \le λ\le 1$ and $T >0$. The Riemann-von Mangoldt Theorem confirms this estimate when $λ=1/2$, with $T\spε$ being replaced by $\log T$. In an attempt to transform Backlund's 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 $N(λ, T) \le 8.734 \log T$ for any $1/2 < λ< 1$ and $T\ge 2445999554999$.