[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

It seems that my previous effort to approach S function starting with cubic function was a mistake. I should have started with the linear function first, which was skipped previously. In that case, the cubic curve I found in the plot was too  attractive.

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.

This is where linear approximation is done first

At high imaginary part, the real part of S function is very close to a straight line with a negative slope.

https://www.wolframalpha.com/input?i=plot+re+%28log%28zeta%280.5%2B+d%2B9999i%29%2Fzeta+%280.5-+d%2B9999i%29%29%29++%2Bi+%28-d+log%289999%2F2%2Fpi%29+%29from+-1+to+1
This is where they are overlaid on the same plot.

https://www.wolframalpha.com/input?i=plot+re+%28log%28zeta%280.5%2B+d%2B9999i%29%2Fzeta+%280.5-+d%2B9999i%29%29%29++-%28-d+log%289999%2F2%2Fpi%29+%29from+-1+to+1
And this is where the difference between them is plotted.

https://www.wolframalpha.com/input?i=plot+re+%28log%28zeta%280.5%2B+d%2B9i%29%2Fzeta+%280.5-+d%2B9i%29%29%29++-%28-d+log%289%2F2%2Fpi%29+%29from+-1+to+1
This is where the imaginary part is lower.
https://www.wolframalpha.com/input?i=plot+re+%28log%28zeta%280.5%2B+d%2B4i%29%2Fzeta+%280.5-+d%2B4i%29%29%29++-%28-d+log%284%2F2%2Fpi%29+%29from+-1+to+1
It still applies when the imaginary part is lower than 2 pi.

The shape of the curve doesn't seem to change even when a non-trivial zero is involved.
https://www.wolframalpha.com/input?i=plot+re+%28log%28zeta%280.5%2B+d%2B14.134725142i%29%2Fzeta+%280.5-+d%2B14.134725142i%29%29%29++-%28-d+log%2814.134725142%2F%282+pi%29%29+%29from+-1+to+1

[hamdani yusuf (OP)]

Why 4d geometry makes me sad

The artwork at the end is by Kurt Bruns

Thanks to Daniel Kim for sharing the first two puzzles with me. He mentioned the earliest reference he knows for the tile puzzles is David and Tomei's AMM article titled "The problem of Calissons."

The idea to include the tetrahedron volume example was based on a conversation with Po Shen Lo about these puzzles, during which he mentioned the case of one dimension lower.

I received the cone correction to the proof of Monge's theorem from Akos Zahorsky via email. Also, the Bulgarian team leader Velian Velikov brought up the same argument, and just shot me a message saying "I came across it in a book I found online titled 'Mathematical Puzzles' by Peter Winkler. There, it is attributed to Nathan Bowler"

I referenced quaternions at the end, and if you're curious to learn more, here are a few options.

Also, one of the coolest projects I've ever done was a collaboration with Ben Eater to make interactive videos based on that topic:
https://eater.net/quaternions

Timestamps
0:00 - Intro
0:32 - Twirling tiles
6:45 - Tarski Plank Problem
10:24 - Monge?s Theorem
17:26 - 3D Volume, 4D answer
18:51 - The hypercube stack
25:52 - The sadness of higher dimensions

Unfortunately, visualizing Zeta function is best suited in 4 dimensions: 2 for complex input and 2 for complex output.

[hamdani yusuf (OP)]

Riemann Hypothesis 4 : Implications of Anomalous Zeros

The video describes implications of anomalous zeros on Y and S functions, and try to build an argumentation against their existence. #riemannhypothesis

I'd like to hear your opinions on what should be included in my next video.

[hamdani yusuf (OP)]

Here is my next video on Riemann's Hypothesis, which introduces V function.

This video introduces V function, which is a combination between the difference and ratio of zeta function and its analytic continuation.

[hamdani yusuf (OP)]

Here's the summary of the argumentation in the video above.
The formula of zeta function and its analytic continuation are completely transparent.
There is no anomalous zero with real part >1, according to converging infinite sum formula of Zeta function.
Analytic continuation of zeta function through functional equation implies that there is no anomalous zero with real part <0.
Numerical check found no anomalous zero with positive imaginary part <1 million.
The shapes of V function which have no irregularity, even when non-trivial zero is involved.
The regularity of V function is assumed to include the points where the anomalous zeros of zeta function reside.
The existence of anomalous zeros of zeta function will have profound implications on the shape of V function. There's only two possibilities:
1. The shape of V function still follows regular pattern.
2. The shape of V function becomes irregular at that point of imaginary part.

[hamdani yusuf (OP)]

The existence of anomalous zeros of zeta function will have profound implications on the shape of V function. There's only two possibilities:
1. The shape of V function still follows regular pattern.
2. The shape of V function becomes irregular at that point of imaginary part.

In first case, the shape of the real part of Y function and S function become a flat horizontal line. It's at negative infinity in Y function, and it's at 0 in S function. This is impossible, considering the formula for zeta function.