[hamdani yusuf (OP)]

Here's the plot from 0 to i
https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%287%2B+t+i%29+%2F%28Zeta%281-7%2B+t+i%29%29%29%29-re%28log%28zeta%286%2B+t+i%29+%2F%28Zeta%281-6%2B+t+i%29%29%29%29%2F2-re%28log%28zeta%288%2B+t+i%29+%2F%28Zeta%281-8%2B+t+i%29%29%29%29%2F2+from+0+to+1

Between 0 and 10i
https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%287%2B+t+i%29+%2F%28Zeta%281-7%2B+t+i%29%29%29%29-re%28log%28zeta%286%2B+t+i%29+%2F%28Zeta%281-6%2B+t+i%29%29%29%29%2F2-re%28log%28zeta%288%2B+t+i%29+%2F%28Zeta%281-8%2B+t+i%29%29%29%29%2F2+from+0+to+10
At around i1.8 the curve seems to bend sharper than elsewhere.

[hamdani yusuf (OP)]

Here's the plot without log
https://www.wolframalpha.com/input?i=plot+%28zeta%287%2B+t+i%29+%2F%28Zeta%281-7%2B+t+i%29%29%29-%28zeta%286%2B+t+i%29+%2F%28Zeta%281-6%2B+t+i%29%29%29%2F2-%28zeta%288%2B+t+i%29+%2F%28Zeta%281-8%2B+t+i%29%29%29%2F2+from+0+to+10

https://www.wolframalpha.com/input?i=plot+re%28%28zeta%287%2B+t+i%29+%2F%28Zeta%281-7%2B+t+i%29%29%29-%28zeta%286%2B+t+i%29+%2F%28Zeta%281-6%2B+t+i%29%29%29%2F2-%28zeta%288%2B+t+i%29+%2F%28Zeta%281-8%2B+t+i%29%29%29%2F2%29+from+0+to+2

https://www.wolframalpha.com/input?i=plot+re%28%28zeta%287%2B+t+i%29+%2F%28Zeta%281-7%2B+t+i%29%29%29-%28zeta%286%2B+t+i%29+%2F%28Zeta%281-6%2B+t+i%29%29%29%2F2-%28zeta%288%2B+t+i%29+%2F%28Zeta%281-8%2B+t+i%29%29%29%2F2%29+from+0+to+10

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+log%28%28zeta%287%2B+t+i%29+%2F%28Zeta%281-7%2B+t+i%29%29%29-%28zeta%286%2B+t+i%29+%2F%28Zeta%281-6%2B+t+i%29%29%29%2F2-%28zeta%288%2B+t+i%29+%2F%28Zeta%281-8%2B+t+i%29%29%29%2F2%29+from+0+to+10
This shows how the height of the spike evolves from low to high imaginary part.

The higher imaginary part, the smaller the spikes become.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%287%2B+t+i%29+%2F%28Zeta%281-7%2B+t+i%29%29%29-%28zeta%286%2B+t+i%29+%2F%28Zeta%281-6%2B+t+i%29%29%29%2F2-%28zeta%288%2B+t+i%29+%2F%28Zeta%281-8%2B+t+i%29%29%29%2F2%29+%29+from+100+to+110

[hamdani yusuf (OP)]

Let's apply the spike detector around the critical line.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%280.5%2B+t+i%29+%2F%28Zeta%281-0.5%2B+t+i%29%29%29-%28zeta%280.49%2B+t+i%29+%2F%28Zeta%281-0.49%2B+t+i%29%29%29%2F2-%28zeta%280.51%2B+t+i%29+%2F%28Zeta%281-0.51%2B+t+i%29%29%29%2F2%29+%29+from+-5+to+30

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%280.5%2B+t+i%29+%2F%28Zeta%281-0.5%2B+t+i%29%29%29-%28zeta%280.4%2B+t+i%29+%2F%28Zeta%281-0.4%2B+t+i%29%29%29%2F2-%28zeta%280.6%2B+t+i%29+%2F%28Zeta%281-0.6%2B+t+i%29%29%29%2F2%29+%29+from+0+to+30

[hamdani yusuf (OP)]

This is also a spike detector https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%280.5%2B+t+i%29+%2F%28Zeta%281-0.5%2B+t+i%29%29%29-re%28log%28zeta%280.4%2B+t+i%29+%2F%28Zeta%281-0.4%2B+t+i%29%29%29%29%2F2-re%28log%28zeta%280.6%2B+t+i%29+%2F%28Zeta%281-0.6%2B+t+i%29%29%29%29%2F2%29+from+0+to+30

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%280.5%2B+t+i%29+%2F%28Zeta%281-0.5%2B+t+i%29%29%29-re%28log%28zeta%280.49%2B+t+i%29+%2F%28Zeta%281-0.49%2B+t+i%29%29%29%29%2F2-re%28log%28zeta%280.51%2B+t+i%29+%2F%28Zeta%281-0.51%2B+t+i%29%29%29%29%2F2%29+from+0+to+30

These show the precision limit of Wolfram alpha. It should be 0 due to symmetrysymmetry of the S function about the critical line.

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%280.49999%2B+s+i%29%29+%2F+%28Zeta%281-0.49999+%2B+s+i%29%29%29%29+from+-9+to+9

Plotted symmetrically, it looks like the letter V. So, I'll just call it V function.

At a glance the right hand side looks like a logarithmic function. But this is how it looks when the log is dropped.
https://www.wolframalpha.com/input?i=plot++%28abs%28%28zeta%280.49%2B+i+s%29%29+%2F+%28Zeta%281-0.49+%2B+i+s+%29%29%29%29+from+-9+to+9

When it's exponentiated.
https://www.wolframalpha.com/input?i=plot+e%5E%28abs%28%28zeta%280.49%2B+i+s%29%29+%2F+%28Zeta%281-0.49+%2B+i+s+%29%29%29%29+from+-9+to+9
And doubly exponentiated.
https://www.wolframalpha.com/input?i=plot+e%5Ee%5E%28abs%28%28zeta%280.49%2B+i+s%29%29+%2F+%28Zeta%281-0.49+%2B+i+s+%29%29%29%29+from+-9+to+9
When s is replaced by e^s, we get a curve resembling a linear function on the right hand side.
https://www.wolframalpha.com/input?i=plot++%28abs%28%28zeta%280.49%2B+i+e%5Es%29%29+%2F+%28Zeta%281-0.49+%2B+i+e%5Es+%29%29%29%29+from+-9+to+9

[hamdani yusuf (OP)]

When absolute value of s is used.
https://www.wolframalpha.com/input?i=plot++%28abs%28%28zeta%280.49%2B+i+e%5E%7Cs%7C%29%29+%2F+%28Zeta%281-0.49+%2B+i+e%5E%7Cs%7C+%29%29%29%29+from+-9+to+9

https://www.wolframalpha.com/input?i=plot++%28abs%28%28zeta%280.49%2B+i+e%5E%7Cs%7C%29%29+%2F+%28Zeta%281-0.49+%2B+i+e%5E%7Cs%7C+%29%29%29%29+from+-1+to+1

https://www.wolframalpha.com/input?i=plot++%28abs%28%28zeta%280.49%2B+i+e%5E%7Cs%7C%29%29+%2F+%28Zeta%281-0.49+%2B+i+e%5E%7Cs%7C+%29%29%29%29+from+-0.1+to+0.1

[hamdani yusuf (OP)]

It turns out that Zeta (2 pi i) isn't exactly the same as Zeta (1+2 pi i).

This plot shows where they become the same.
https://www.wolframalpha.com/input?i=plot+re%28log%28+zeta%28i+x%29+%2Fzeta%281%2Bi+x%29+%29+%29+from+2+pi+to+2.00000002+pi

It's close to 2.00000001 π

[hamdani yusuf (OP)]

A 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%

https://www.wolframalpha.com/input?i=plot+re%28log%28+zeta%280.499999+%2B+i+x%29+%2Fzeta%281-0.499999%2Bi+x%29+%29+%29++from+2+pi+to+2.004+pi
While the inflection point seems to be close to 2.002 π

In these plots, it seems like we are getting close to the precision limit of Wolfram alpha.
https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%280.499999%2Bi+x%29%2Fzeta%281-0.499999%2Bi+x%29%29%29+from+2.0021169+pi+to+2.0021171+pi

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%280.499999%2Bi+x%29%2Fzeta%281-0.499999%2Bi+x%29%29%29+from+2.00211695+pi+to+2.0021170+pi

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%280.49999999%2Bi+x%29%2Fzeta%281-0.49999999%2Bi+x%29%29%29+from+2.00211695+pi+to+2.0021171+pi

[hamdani yusuf (OP)]

It turns out that Zeta (2 pi i) isn't exactly the same as Zeta (1+2 pi i).

This plot shows where they become the same.
https://www.wolframalpha.com/input?i=plot+re%28log%28+zeta%28i+x%29+%2Fzeta%281%2Bi+x%29+%29+%29+from+2+pi+to+2.00000002+pi

It's close to 2.00000001 π

The plot below shows where the S function produces zero as output where the real part of input deviates from the critical line.

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%280.5-10%5E-1%2Bi+x%29%2Fzeta%280.5%2B10%5E-1%2Bi+x%29%29%29+from+2.002032256+pi+to++2.00203229+pi

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%280.5-10%5E-4%2Bi+x%29%2Fzeta%280.5%2B10%5E-4%2Bi+x%29%29%29+from+2.002116+pi+to+2.0021179+pi

And manually make the plot.

[hamdani yusuf (OP)]

y in the table above represents the imaginary part of S function that produce a curve that crosses zero at positions +/- x distance away from critical line.

[hamdani yusuf (OP)]

The plot below shows where the S function produces zero as output where the real part of input deviates from the critical line.

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%280.5-10%5E-1%2Bi+x%29%2Fzeta%280.5%2B10%5E-1%2Bi+x%29%29%29+from+2.002032256+pi+to++2.00203229+pi

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%280.5-10%5E-4%2Bi+x%29%2Fzeta%280.5%2B10%5E-4%2Bi+x%29%29%29+from+2.002116+pi+to+2.0021179+pi

And manually make the plot.

It turns out that the curve I obtained from visual approximation is very close to a quadratic equation.

[hamdani yusuf (OP)]

When the negative x is included, we get a nice parabolic curve.



This shows that when the imaginary part its input is above i(2+β)π, which is the inflection limit, S function can not produce 0 when the input has non-zero distance from the critical line. β≈0.002117...
Consequently, Riemann's Zeta function cannot produce zero out of critical line when the imaginary part its input is above i(2+β)π.

[hamdani yusuf (OP)]

We get a nice full wave when the imaginary part is exactly 2*pi

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+0+to+1

But later on, we found out that S function  doesn't produce 0 at exactly 2πi.

It turns out that Zeta (2 pi i) isn't exactly the same as Zeta (1+2 pi i).

This plot shows where they become the same.
https://www.wolframalpha.com/input?i=plot+re%28log%28+zeta%28i+x%29+%2Fzeta%281%2Bi+x%29+%29+%29+from+2+pi+to+2.00000002+pi

It's close to 2.00000001 π

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+2.00000001+pi+i%29+%2F%28Zeta%281-x%2B+2.00000001+pi+i%29%29%29%29from+0+to+1

[hamdani yusuf (OP)]

It looks like a cubic equation with roots at 0, 0.5, and 1.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.00000001+pi+i%29%29+%2F+%28Zeta%281-x%2B2.00000001+pi+i%29%29%29%29+-i%2F235.96+%28x%29%28x-0.5%29%28x-1%29+from+0+to+1

But when zoomed out, it's different.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.00000001+pi+i%29%29+%2F+%28Zeta%281-x%2B2.00000001+pi+i%29%29%29%29+-i%2F235.96+%28x%29%28x-0.5%29%28x-1%29+from+-5+to+6

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002117+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002117+pi+i%29%29%29%29+-+i%2F236.31+%28x-0.5%29%5E3+from+0+to+1
The inflection point looks like a cubic equation.

Here's when zoomed in around imaginary axis
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002117+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002117+pi+i%29%29%29%29+-+i%2F236.31+%28x-0.5%29%5E3+from+0+to+0.001

and around critical line
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002117+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002117+pi+i%29%29%29%29+-+i%2F236.31+%28x-0.5%29%5E3+from+0.45+to+0.55

But when zoomed out, it's still different.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002117+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002117+pi+i%29%29%29%29+-+i%2F236.31+%28x-0.5%29%5E3+from+-5+to+6

[hamdani yusuf (OP)]

This plot shows a more precise value of the inflection point of S function. It's near the precision limit of Wolfram alpha.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.0021169777+pi+i%29%29+%2F+%28Zeta%281-x%2B2.0021169777+pi+i%29%29%29%29+-+i%2F236.31+%28x-0.5%29%5E3+from+0.4999+to+0.5001


Even slightly higher imaginary part leads to an obviously negative slope curve, eliminating the possibility of nontrivial zero outside of the critical line.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.002117+pi+i%29%29+%2F+%28Zeta%281-x%2B2.002117+pi+i%29%29%29%29+-+i%2F236.31+%28x-0.5%29%5E3+from+0.4999+to+0.5001

[hamdani yusuf (OP)]

The study of S function around its inflection point can be crucial for proof of Riemann hypothesis. It can be done by demonstrating that its behavior above the inflection point can be generalized up to infinity.
The aforementioned important behavior is that for any point s on the right side of the critical line above the inflection point in complex plane, its S function is always smaller than S function of its reflection about the critical line. They can't be equally zero, which is contrary to the functional equation for Riemann Zeta function.
Thus nontrivial zero of Riemann Zeta function can't be on the right side nor the left side of the critical line. They must be precisely on the critical line.

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.0021169954+pi+i%29%29+%2F+%28Zeta%281-x%2B2.0021169954+pi+i%29%29%29%29+%2B%281%2F236.31+%28x-0.5%29%5E3%29+%2Bi%2F30400%28x%29%28x-0.5%29%5E3%28x-1%29from+-0+to+1
When the S function at inflection point is "corrected" using a cubic equation, there's still a quintic equation "residue".

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.0021169954+pi+i%29%29+%2F+%28Zeta%281-x%2B2.0021169954+pi+i%29%29%29%29+%2B%281%2F236.31+%28x-0.5%29%5E3%29+-1%2F30400%28x%29%28x-0.5%29%5E3%28x-1%29from+-0.01+to+1.01
Even when both corrections are applied, there's still some residue.

[hamdani yusuf (OP)]

Let's go down the rabbit hole of curve fitting.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.0021169954+pi+i%29%29+%2F+%28Zeta%281-x%2B2.0021169954+pi+i%29%29%29%29+%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+-+i%2F2300000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+0+to+1

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.0021169954+pi+i%29%29+%2F+%28Zeta%281-x%2B2.0021169954+pi+i%29%29%29%29+%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+-+i%2F2150000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+0+to+1

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2.0021169954+pi+i%29%29+%2F+%28Zeta%281-x%2B2.0021169954+pi+i%29%29%29%29+%2B%281%2F236.30994+%28x-0.5%29%5E3%29+-1%2F30391.7%28x%29%28x-0.5%29%5E3%28x-1%29+-+i%2F2480000%28x%29%28x-0.2%29%28x-0.5%29%28x-0.8%29%28x-1%29from+0+to+1