Jacob's ladders, interactions between \(\zeta\)-oscillating systems, and a \(\zeta\)-analogue of an elementary trigonometric identity (Q1661317)

In previous papers, the author has introduced the following notions, within the theory of the Riemann zeta function: Jacob's ladders, oscillating systems, \(\zeta\)-factorization, metamorphoses, and more. In the present paper, the author obtains a \(\zeta\)-analogue of an elementary trigonometric identity and other interactions between oscillating systems. Truly, on reading the paper, one has the impression to enter in a rich and promising world, within the realm of the theory of the Riemann zeta function. In Remark 6, the author defends the opinion that the Riemann-Siegel formula represents Riemann's fundamental contribution to the theory of oscillations (independently from the analytic number theory uses). Namely, that it is the fate of Riemann's oscillations to describe profound laws of our universe. In Remark 16, sequences behaving like one-dimensional Friedmann-Hubble universes are specified. Anyhow, at this stage, it is not clear to this reviewer the possible explicit application of the expressions in the paper to the comprehension of the cosmos, and we will have to wait for further development of this line of work. It does not help that the author only mentions his own papers and just one extra, by \textit{C. L. Siegel} [Quell. Stud. Gesch. Math. B 2, 45--80 (1932; Zbl 0004.10501)].