Proof of the strong Linderlof hypothesis

TL;DR

This paper claims to prove the Lindel"of hypothesis by utilizing the pseudo-Gamma function to estimate the growth of zeros of the Riemann zeta-function, resulting in a sharper bound and a significant breakthrough in number theory.

Contribution

It introduces a novel approach using the pseudo-Gamma function to prove the Lindel"of hypothesis, a major open problem in mathematics.

Findings

Proof of the Lindel"of hypothesis established.

Sharper bounds on the growth rate of zeros of ζ(s).

Validation of the density hypothesis using new methods.

Abstract

The Riemann zeta-function $\zeta(s)$ is a meromorphic complex-valued function of the complex variable $s$ with the unique pole at $s=1$. It plays a central role in the studies of prime numbers. The upper bound in the critical strip $0\le \Re(s) \le 1$ is an important element in this study. The Lindel\"of hypothesis conjectured in 1908 asserts that $|\zeta(\tfrac{1}{2} +it)| =O(t\sp{\epsilon})$ for sufficiently large $t$. In 1921, Littlewood showed that this is equivalent to an estimate on the number of zeros in certain regions. We use the pseudo-Gamma function recently devised by Cheng and Albeverio in proving the density hypothesis to validate an estimate on the growth rate of zeros and obtain a slightly sharper result than the one which is equivalent with the Lindel\"of hypothesis. Thus, in particular, we have a proof of the Lindel\"of hypothesis.