Zeros of Riemann's Zeta Functions in the Line z=1/2+it(0)

Ovchinnikov, Yury N.

doi:10.1007/s10948-019-05243-0

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Zeros of Riemann's Zeta Functions in the Line z=1/2+it(0)

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Ovchinnikov, Yury N.
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Ovchinnikov, Y. N. (2019). Zeros of Riemann's Zeta Functions in the Line z=1/2+it(0). Journal of Superconductivity and Novel Magnetism, 32(11), 3363-3368. doi:10.1007/s10948-019-05243-0.

Cite as: https://hdl.handle.net/21.11116/0000-0005-8FDE-1

Abstract

It was found that, in addition to trivial zeros in points (z=-2N,N=1, 2..., natural numbers), the Riemann's zeta function zeta (z) has zeros only on the line {z=<mml:mfrac>12</mml:mfrac>+it0, t(0) is real}. All zeros are numerated, and for each number, N, the positions of the non-overlap intervals with one zero inside are found. The simple equation for the determination of centers of intervals is obtained. The analytical function eta (z), leading to the possibility fix the zeros of the zeta function zeta (z), was estimated. To perform the analysis, the well-known phenomenon, phase-slip events, is used. This phenomenon is the key ingredient for the investigation of dynamical processes in solid-state physics, for example, if we are trying to solve the TDGLE (time-dependent Ginzburg-Landau equation).