[hamdani yusuf (OP)]

In real life, we can often get useful information without knowing the absolute value of something. The difference or ratio between two or more related things are sometimes enough for us to solve our problems at hand.
That's what the functions I introduced previously are offering to prove Riemann's hypothesis.

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+s+%2B+10+sin%282+s%29+from+-20+to+20

https://www.wolframalpha.com/input?i=plot+s+%2B+10+sin%284+s%29%2Fs+from+-20+to+20

https://www.wolframalpha.com/input?i=plot+s+%2B+10+sin%28tan%28s%29%29+from+-20+to+20

These are some other analogies to describe the problem. The functions are combination between a simple pattern and a more complex pattern. But we can see that at high input value,  the plot never crosses 0 again, because the simple pattern greatly outweighs the more complex pattern.

[hamdani yusuf (OP)]

After watching a video about elliptic curve, I suspected that trivial zeros of Zeta function must somehow be related to non-trivial zeros. This video shows how they are related.

The video shows that non-trivial zeros of Riemann's Zeta function must somehow be correlated to its trivial zeros. I think it's worth exploring if other points on the critical line are also correlated to points on the real line through some sort of mapping or projection.

Imagine that smallest non-trivial zero, around (0.5+14.13i) is corresponding to smallest negative even integer, ie (-2,0i). Larger non-trivial zeros are corresponding to larger negative even integers. Every non-trivial zero exists where S function has negative slope. That's when |im(s)| > (2+β) π i, which is the inflection point of S function. The role of that point on critical line is likely corresponding to (0, 0) for real line, as the edge of critical strip closest to trivial zeros.

[hamdani yusuf (OP)]

In transition zone between 2 π i and (2+β) πi, S function crosses zero more than once. It is corresponding to critical strip. Below 2 π i, S function in critical strip has positive slope. It's corresponding to real line >1. Critical line between 2 π i and 0 is corresponding to real line between 1 and +~.
This correlation is interesting because e^(2 π i) = 1.

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.

The behavior of S function around its inflection point reminds me of Riemann sphere.

https://en.wikipedia.org/wiki/Riemann_sphere
In 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.

Here's an online simulator.
https://www.geogebra.org/m/gD7Rygd2

[hamdani yusuf (OP)]

Let's recap the proof or Riemann's hypothesis using S function.
For every non-trivial zero, ζ(s) = ζ(1-s*) = 0.
S function at that point must be 0.
Results of S function can be divided into three zones.
First is where S function inside the critical strip has positive slope as it crosses zero. It spans from 0i to 2πi.
Second is where S function inside the critical strip has both positive and negative slope as it crosses zero. It spans from 2πi to (2+β)πi.
Third is where S function inside the critical strip has negative slope as it crosses zero. It spans from (2+β)πi to ∞i.
In the first and third zones, the curve only crosses zero exactly once, ie at re(s) =1/2. While in the second zone, it crosses zero three times, ie at re(s) =1/2 and 2 others which are symmetrical about the critical line.
Violation of Riemann's hypothesis implies that S function can cross zero more than once in the third zone, which is contradicting the observations.

[hamdani yusuf (OP)]

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

Outside of critical strip, it looks more like a letter V.
https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%28-15.5%2B+s+i%29%29+%2F+%28Zeta%281%2B15.5%2B+s+i%29%29%29%29+from+-9+to+9

[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

This curve is very similar to a cubic equation 1/1887.68 * (2x-1)(1-(2x-1)^2).
Here's the plot of both curves in the same graph (the cubic equation is multiplied by i).

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29+%2B+i%2F1887.68+%282x-1%29%281-%282x-1%29%5E2%29+from+0+to+1

Although when zoomed in, there is still visible difference.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29+%2Bi%2F1887.68+%282x-1%29%281-%282x-1%29%5E2%29+from+0.784+to+0.793

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29+-+1%2F1887.68++%282x-1%29%281-%282x-1%29%5E2%29+from+0+to+1
The difference between S function and the cubic equation looks like a quintic equation.

At a glance, it looks like this
https://www.wolframalpha.com/input?i=plot+%282x-1%29%281-%282x-1%29%5E2%29+%281-%281.7%282x-1%29%29%5E2%29from+0+to+1

[hamdani yusuf (OP)]

A more precise constants can be obtained by showing both curves in the same plot.
https://www.wolframalpha.com/input?i=plot+3.38+i++10%5E-7%282x-1%29%281-%282x-1%29%5E2%29+%281-%281.727%282x-1%29%29%5E2%29+%2B+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29+-+1%2F1887.68++%282x-1%29%281-%282x-1%29%5E2%29+from+0+to+1

And we can plot the difference.
https://www.wolframalpha.com/input?i=plot+-3.38+++10%5E-7%282x-1%29%281-%282x-1%29%5E2%29+%281-%281.727%282x-1%29%29%5E2%29+%2B+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29+-+1%2F1887.68++%282x-1%29%281-%282x-1%29%5E2%29+from+0+to+1

Slightly changing a constant can reduce the number of peaks.
https://www.wolframalpha.com/input?i=plot+-3.41+10%5E-7%282x-1%29%281-%282x-1%29%5E2%29+%281-%281.727%282x-1%29%29%5E2%29+%2B+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29+-+1%2F1887.68++%282x-1%29%281-%282x-1%29%5E2%29+from+0+to+1

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%28x%2Bi%29%29+-+%28Zeta%28x-i%29%29%29%29from+-20+to+21
The plot shows the log of difference between ζ(s) and its complex conjugate.

https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%28x%2B0.001i%29%29+-+%28Zeta%28x-0.001i%29%29%29%29from+-10+to+2
And this is for smaller imaginary parts.

https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%28x%2Bi%29%29+%2F+%28Zeta%28x-i%29%29%29%29from+-20+to+21
While this one shows the ratio

https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%28x%2Bi%29%29+%2F+%28Zeta%28x-i%29%29%29%29from+-20+to+2
This one is more focused on the negative real of s.

https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%28x%2B0.001i%29%29+%2F+%28Zeta%28x-0.001i%29%29%29%29from+-10+to+5
And this is for smaller imaginary parts, which reveals where the trivial zeros are, ie negative even integers.

[hamdani yusuf (OP)]

Some geometry behind the Basel problem

This Basel problem is a special case of Zeta function with historical significance. The video shows how to solve it by involving some geometry. It also involves some steps which seem to come out of nowhere, and only becomes clear after further calculations got the problem simplified later on.

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%28x%2B0.001i%29%29+%2F+%28Zeta%28x-0.001i%29%29%29%29from+-10+to+5
And this is for smaller imaginary parts, which reveals where the trivial zeros are, ie negative integers

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B0.001i%29%29+%2F+%28Zeta%28x-0.001i%29%29%29%29from+-10+to+5

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B10.00i%29%29+%2F+%28Zeta%28x-10.00i%29%29%29%29from+-10+to+5

When only the real part is shown, the plot seems to have exceeded the precision limit of Wolfram Alpha.

[hamdani yusuf (OP)]

As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0.

Example.
The Maclaurin series of the exponential function e^x is

https://en.m.wikipedia.org/wiki/Taylor_series

[hamdani yusuf (OP)]

I'm pretty sure that Riemann Hypothesis can be proven by expressing S function, especially at high imaginary part, in terms of Taylor-MaClaurin series. It will demonstrate that S function can only be zero at critical line, while the imaginary part of its input is higher than (2+β) π i.

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

This curve is very similar to a cubic equation 1/1887.68 * (2x-1)(1-(2x-1)^2).
Here's the plot of both curves in the same graph (the cubic equation is multiplied by i).

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29+%2B+i%2F1887.68+%282x-1%29%281-%282x-1%29%5E2%29+from+0+to+1

Although when zoomed in, there is still visible difference.
https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29+%2Bi%2F1887.68+%282x-1%29%281-%282x-1%29%5E2%29+from+0.784+to+0.793

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2+pi+i%29%29+%2F+%28Zeta%281-x%2B2+pi+i%29%29%29%29+-+1%2F1887.68++%282x-1%29%281-%282x-1%29%5E2%29+from+0+to+1
The difference between S function and the cubic equation looks like a quintic equation.

At a glance, it looks like this
https://www.wolframalpha.com/input?i=plot+%282x-1%29%281-%282x-1%29%5E2%29+%281-%281.7%282x-1%29%29%5E2%29from+0+to+1

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+%28log%28zeta%28x%2B999i%29+%2F%28Zeta%281-x%2B999+i%29%29%29%29from+0+to+1

At higher imaginary part, the first degree (a straight line) overwhelmingly dominate the result. It implies that S function can only be zero at critical line, and proves that Riemann hypothesis is true.

[hamdani yusuf (OP)]

This curve is very similar to a cubic equation 1/1887.68 * (2x-1)(1-(2x-1)^2).

In expanded form, it's
-0.002119 x + 0.00635701 x^2 - 0.00423801 x^3

While the quintic equation is
0.000032259 x^5 - 0.0000806476 x^4 + 0.0000698788 x^3 - 0.0000241707 x^2 + 0.00000268038 x

[hamdani yusuf (OP)]

Let's observe how S function progresses from 2 pi i down to 0.
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+-20+to+21

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B++pi+i%29+%2F%28Zeta%281-x%2B++pi+i%29%29%29%29from+-20+to+21

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B++pi+i%29+%2F%28Zeta%281-x%2B++pi+i%29%29%29%29from+-20+to+21

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+1%2F2+pi+i%29+%2F%28Zeta%281-x%2B+1%2F2+pi+i%29%29%29%29from+-20+to+21

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+1%2F6+pi+i%29+%2F%28Zeta%281-x%2B+1%2F6+pi+i%29%29%29%29from+-20+to+21

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+1%2F60+pi+i%29+%2F%28Zeta%281-x%2B+1%2F60+pi+i%29%29%29%29from+-20+to+21

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0+i%29+%2F%28Zeta%281-x%2B+0+i%29%29%29%29from+-20+to+21

When the imaginary part is not omitted:
https://www.wolframalpha.com/input?i=plot+%28log%28zeta%28x%2B+0.0+i%29+%2F%28Zeta%281-x%2B+0.0+i%29%29%29%29from+-20+to+21

At high value of x the program seem to exceed its operability range.
https://www.wolframalpha.com/input?i=plot+%28log%28zeta%28x%2B+0.0+i%29+%2F%28Zeta%281-x%2B+0.0+i%29%29%29%29from+-50+to+51

When Im(s)=0, the function shows infinitely tall spikes down on negative real side, and infinitely tall spikes up on positive real side.
With increased imaginary part, the spikes are quickly reduced and smoothed out
Here's the spike around Re(s)=1.
https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0+i%29+%2F%28Zeta%281-x%2B+0+i%29%29%29%29from+0.5+to+1.5

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0.1+i%29+%2F%28Zeta%281-x%2B+0.1+i%29%29%29%29from+0.5+to+1.5

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0.2+i%29+%2F%28Zeta%281-x%2B+0.2+i%29%29%29%29from+0.5+to+1.5
We're getting close to inflection point.

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0.222892341+i%29+%2F%28Zeta%281-x%2B+0.222892341+i%29%29%29%29from+1.2305+to+1.23075
Here's the more precise inflection point.

[hamdani yusuf (OP)]

This is the same as the last curve, just zoomed out.
https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0.222892341+i%29+%2F%28Zeta%281-x%2B+0.222892341+i%29%29%29%29from+0+to+2

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0.222892341+i%29+%2F%28Zeta%281-x%2B+0.222892341+i%29%29%29%29from+0+to+20

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0.222892341+i%29+%2F%28Zeta%281-x%2B+0.222892341+i%29%29%29%29from+-20+to+21

The peaks are still popping out elsewhere, other than Re(s)=0 and Re(s)=1

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0.222892341+i%29+%2F%28Zeta%281-x%2B+0.222892341+i%29%29%29%29from+4+to+10
Let's focus on the highest peak, around Re(s)=7, and the through preceding it, around Re(s)=6.

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+0.222892341+i%29+%2F%28Zeta%281-x%2B+0.222892341+i%29%29%29%29from+4+to+8

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+1+i%29+%2F%28Zeta%281-x%2B+1+i%29%29%29%29from+4+to+10
increasing the imaginary part to i smooths the curve further, but some indentations are still visible.

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+1.5+i%29+%2F%28Zeta%281-x%2B+1.5+i%29%29%29%29from+4+to+10

https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+2+i%29+%2F%28Zeta%281-x%2B+2+i%29%29%29%29from+4+to+10
at 2i, the indentations can't be seen anymore.

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%28x%2B2++i%29%29+%2F+%28Zeta%281-x%2B2++i%29%29%29%29+-+i%2F300%28x%2B13.5%29+%28x-0.5%29+%28x-14%29++from+-20+to+21
Here's a cubic curve with the same zero overlaid in the same plot. They are quite different elsewhere.