Riemann hypothesis and the arc length of the Riemann $Z(t)$-curve

TL;DR

This paper proves that under the Riemann hypothesis, the arc length of the Riemann Z-curve asymptotically equals twice the sum of local maxima of Z(t), with additional insights from Jacob's ladders.

Contribution

It establishes a new asymptotic relation between the arc length of the Z-curve and local maxima, incorporating Jacob's ladders into the analysis.

Findings

Arc length asymptotically equals double the sum of local maxima

Provides a new integral involving Jacob's ladders

Enhances understanding of Z(t) behavior under Riemann hypothesis

Abstract

On Riemann hypothesis it is proved in this paper that the arc length of the Riemann $Z$-curve is asymptotically equal to the double sum of local maxima of the function $Z(t)$ on corresponding segment. This paper is English remake of our paper \cite{9}, with short appendix concerning new integral generated by Jacob's ladders added.