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Quote from: hamdani yusuf on 17/05/2024 05:14:38We get a nice full wave when the imaginary part is exactly 2*pihttps://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+2+pi+i%29+%2F%28Zeta%281-x%2B+2+pi+i%29%29%29%29from+0+to+1The same curve looks like an inflection when zoomed out. https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+2+pi+i%29+%2F%28Zeta%281-x%2B+2+pi+i%29%29%29%29from+-10+to+11A more accurate inflection curve has a slightly bigger imaginary number. https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+2.002117+pi+i%29+%2F%28Zeta%281-x%2B+2.002117+pi+i%29%29%29%29from+0.495+to+0.505
We get a nice full wave when the imaginary part is exactly 2*pi
Exploration of backslash function (aka S function) around its inflection point can be exciting in its own right, but does not have much effect on the determination of Riemann hypothesis, which for now has narrowed down to critical strip with extremely high imaginary part.
https://en.wikipedia.org/wiki/Riemann_sphereIn mathematics, the Riemann sphere, named after Bernhard Riemann,[1] is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point ∞ is near to very large numbers, just as the point 0 is near to very small numbers.
Let's observe how S function progresses from 2 pi i down to 0.