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 Lindelöf hypothesis conjectured in 1908 asserts that $|ζ(\tfrac{1}{2} +it)| =O(t\spε)$ 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 Lindelöf hypothesis. Thus, in particular, we have a proof of the Lindelöf hypothesis.

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References in corpus (2)

• A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis
• Proof of the the Riemann hypothesis from the density and Lindelof hypotheses via a power sum method