# Proof of the strong Density Hypothesis

Yuanyou Cheng

Arxiv ID: 0810.2103•Last updated: 6/16/2021

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 $N(\lambda, T) =O\bigl(T\sp{2(1-\lambda)
+\epsilon} \bigr)$ for any $\epsilon >0$, where $N(\lambda, T)$ is the number
of zeros of $\zeta(s)$ when $\Re(s) \ge\lambda$ 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 $T\sp{\epsilon}$ 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
$N(\lambda, T) \le 8.734 \log T$ for any $1/2 < \lambda < 1$ and $T\ge
2445999554999$.

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