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**New Theories / Re: What makes Riemann's Hypothesis Hard to Prove?**

« **on:**17/05/2024 04:33:18 »

Y function convincingly shows the symmetry of Riemann's Zeta function around the critical line. It also shows the increasing stability for higher imaginary part. But somehow some of us aren't convinced that all non-trivial zeros of Riemann's Zeta function are on the critical line.Y function guarantees that at the critical line, its output is always negative infinity, because on that line, s=1-s*. Consequently,

Y(s) = Ln (ζ(s)-ζ(s)) = Ln(0) = -~

To prove Riemann's hypothesis, a further step is required. It needs to show that ζ(s)<>ζ(1-s*) when s<>1-s*.

This is where the backslash function can help.

B(s) = Ln (ζ(s) / ζ(1-s*)) = Ln(ζ(s)) - Ln(ζ(1-s*))

https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%28x%2B20i%29%29+%2F+%28Zeta%281-x%2B20i%29%29%29%29from+-14+to+15

In this case, we only need to consider the value in the critical strip,

https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%28x%2B20i%29%29+%2F+%28Zeta%281-x%2B20i%29%29%29%29from+0+to+1