Proof of the strong Linderlof hypothesis

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.