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Schleich, W.P., Bezdekova, I., Kim, M.B., Abbott, P.C., Maler, H., Montgomery, H.L. and Neuberger, J.W. (2018) Equivalent Formations of the Riemann Hypothesis Based on Lines of Constant Phase. Physica Scripta, 93, Article No. 065201.
https://doi.org/10.1088/1402-4896/aabca9

has been cited by the following article:

• TITLE: Geometric Proof of Riemann Conjecture (Continued)

  AUTHORS: Chuanmiao Chen

  KEYWORDS: RC, Equivalence, RC2, Product Expression, Single Peak, Multiple Zeros

  JOURNAL NAME: Advances in Pure Mathematics, Vol.11 No.9, September 22, 2021

  ABSTRACT: This paper will prove Riemann conjecture(RC): All zeros of ξ(τ) lie on critical line. Denote ,and on critical line. We have found two mysteries in Riemann’s paper. The first mystery is the equivalence: is uniquely determined by its initial value u (t). The second mystery is Riemamm conjecture 2 (RC2): Using all zeros tj of u (t) can uniquely express .We find that the proof of RC is hidden in it. Our basic idea as follows. Consider functional equation . It is known that on critical line and , then we have the upper bound of growth To prove RC2 (or RC), by contradiction. If ξ(τ) has conjugate complex roots t'±iβ'’, β'>0, R2=t'2+β'2, by symmetry ξ(τ)=ξ(-τ), then -(t'±iβ'') do yet. So ξ must contain four factors. Then u(t) contains a real factor and ln|u(t)| contains a term (the lower bound) which contradicts to the growth above. So ξ can not have the complex roots and u(t) does not have the factor p(t). Therefore both RC2 and RC are proved. We have seen that the two-dimensional problem is reduced to one-dimension and the one-dimensional u(t) is reduced to its product expression. Perhaps this is close to the original idea of Riemann. Other results are also discussed by geometric analysis in the last section.