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.
A criterion related to the Riemann Hypothesis
Jacob's ladders, new properties of the function $\arg\zf$ and corresponding metamorphoses
Jacob's ladders and some new consequences from A. Selberg's formula
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