Proof of the Density Hypothesis

Details Proof of the Density Hypothesis Proof of the Density Hypothesis

2008-10-12

arxiv.org/abs/0810.2103v10

Mathematics

General Mathematics

Revised version on 03-08-2010 (17 pages)

Scientific paper

The Riemann zeta function is a meromorphic function on the whole complex plane. It has infinitely many zeros and a unique pole at $s = 1$. Those zeros at $s = -2, -4, -6, ...$ are known as trivial zeros. The Riemann hypothesis, conjectured by Bernhard Riemann in 1859, claims that all nontrivial zeros of $\zeta(s)$ lie on the line $\Re(s) =\tfrac{1}{2}$. 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 $\tfrac{1}{2} \le \lambda \le 1$ and $T >0$. The Riemann-von Mangoldt Theorem confirms this estimate when $\lambda =\tfrac{1}{2}$, with $T\sp{\epsilon}$ being replaced by $\log T$. The xi-function $\xi(s)$ is an entire function involving $\zeta(s)$ and the Euler Gamma function $\Gamma(s)$. This function is symmetric with respect to the line $\Re(s) =\tfrac{1}{2}$, although neither $\zeta(s)$ nor $\Gamma(s)$ exhibits this property. In an attempt to transform Backlund's proof of the Riemann-von Mangoldt Theorem, from 1918, to a proof of the density hypothesis by convexity, we discovered a slightly different approach utilizing a {\it pseudo Gamma function}. This function is symmetric with respect to $\Re(s)=\tfrac{1}{2}$. It is about the size of the Euler Gamma function. Moreover, it is analytic and does not have any zeros and poles in the concerned open region. Aided by this function, we are able to establish a proof of the density hypothesis. Actually, our result is even stronger, when $\tfrac{1}{2} < \lambda < 1$, $N(\lambda, T) =O(\log T)$.

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