24. Eighty-nine
The 3D program that Incheol installed was really helpful.
I was able to draw the Riemann zeta function directly in 3D space on the complex plane, but looking at the Riemann zeta function drawn like this… An interesting idea came to mind.
While the computer was performing function value calculations, I prepared for statistical analysis.
However, the noise that the 486 computer makes is enormous.
I had to run the computer all night to calculate the repeated function values, and it was hard to sleep because of the fan noise.
Riemann zeta function values were calculated over a week, and the results of statistical analysis were applied to some algebraic theories.
Also… !
The next day, I called Professor Park Bo-soo, and I was able to make an appointment with the professor.
Two days later, Korea University.
I put the prepared diskette into Professor Park’s computer, and when I entered the command, the prepared program started.
Created a program name.
by name… .
‘RZF_plotter’
It means Riemann zeta function plotter, and RZF_plotter plots the distribution of Riemann zeta function values in three dimensions in a specified section.
Before running the program, I defined the real range from 0.3 to 0.7 including the critical line, and defined the imaginary range from 0 to +50i, the origin, and then executed the EXE file.
I said to Professor Park Soo-soo.
“Now you have to wait a long time for the graph to be completed.”
Professor Park said with a still questionable expression.
“So… You mean you can see the shape of the Riemann zeta function along the critical line?”
I nodded.
“How accurate is it?”
“In the calculation of the water supply… At points within a radius of 10 from the origin, up to the 500th term is included, and in other areas, the calculation stops at the 100th term. So the accuracy will be pretty high. Instead, it takes a long time to calculate.”
“Even if it were to calculate the series, it would not have been easy to calculate the integral value… .”
“It was very helpful to study numerical analysis in the past.”
Professor Park said with a surprised expression.
“Have you ever studied numerical analysis?”
“yes… I learned a little before.”
I frowned while checking my computer monitor, and Professor Park nodded and said.
“It’s nice though. You don’t come here a lot these days… I wondered if he had lost interest in the Riemann hypothesis.”
I shook my head.
“no. It’s just that the pace is slow… Studying the Riemann hypothesis is still fun.”
The professor nodded with reassurance, and I continued speaking.
“Aside from sleeping and talking to people, there are many times when I actually think about the Riemann hypothesis all day. Some days, even in my dreams, I cling to the Riemann hypothesis.”
“ha ha ha—! Is it that much fun?”
When I nodded, the professor laughed for a while and then said to me.
“I think we should see each other more often. Every time I meet you like this, so that I can be stimulated. ha ha ha–!”
During this conversation, the executed program was terminated.
I opened the newly created file, and there were countless numbers left.
It is a function value calculated at each coordinate of the Riemann zeta function.
I ran the second program.
If the function value of the Riemann zeta function is calculated in the section entered by the program that was started a while ago, the second is to draw the calculated function value in 3D space, and this time the result came out quickly on the monitor.
Professor Park’s eyes widened as he looked at the monitor.
It’s a fact I’ve been feeling for a long time, but… .
Professor Park Soo-Soo’s eyes aren’t small, but maybe he’s just keeping his eyes small?
The professor spoke as if talking to himself.
“The zero position is also accurate… .”
There are a total of 10 zero points between the origin and +50i.
These zero points are around 14i, 21i, 25i, 30i, 33i, 37i, 41i, 43i, 48i, 49i .
“I saw this painting at an international conference last year… this on my computer… I didn’t expect to see it in person like this.”
The professor turned his head to me and continued speaking.
“Now the imaginary interval is up to 50, is there any other interval possible?”
I nodded and said.
“Yes, it is possible. However, it is okay to make the real interval a little larger, but the imaginary interval should not be too long.”
“What do you mean?”
“When I wrote the program, I made the real axis less compact, but since the imaginary axis calculates coordinates in 0.01 units, if the imaginary number section exceeds 500, maybe… The calculation alone will take more than a day.”
The professor nodded at my words.
I opened another file and showed it to the professor.
Again, it was a text file full of numbers.
The professor who checked the file said.
“The numbers here… Are there any zeros?”
The zero points the professor said… .
Exactly, they are the non-obvious zeros that are the core of the Riemann hypothesis.
The number of these non-obvious zeros is infinitely large, all of which are located on a critical line with a real part of 1/2.
“Yes, these are what I found, the imaginary part of a total of 809 zeros.”
“But why are these zero values… ?”
“I just looked at the distance between these zeros.”
The professor nodded as if he knew what he meant.
“Are you talking about the distribution of nuclear energy in quantum mechanics?”
The relationship between the Riemann hypothesis and quantum mechanics was first discovered by Professor Park’s expression of the nuclear energy distribution.
The fact that this expression and the expression representing the distance between the zeros of the Riemann zeta function coincide with each other was discovered by chance in the 1970s, and this finding was numerically verified by a mathematician named Andrew Odlyzko in 1987.
However, my research idea is a little different.
I said to the professor’s question.
“It’s not… It’s a little different… .”
“Another way?”
“Yes, I thought about the link between the Riemann hypothesis and algebra. Do you have a GUE hypothesis?”
The professor nodded.
The Gaussian Unitary Ensemble (GUE) hypothesis.
To put it simply… .
The hypothesis is that there is a statistically significant association between the eigenvalues of a matrix of random numbers and the zeros of the Riemann zeta function.
There is an indicator of the degree of statistical correlation between two samples, which is called covariance.
For example, a covariance of 1 or -1 means a perfectly symmetric relationship between the two samples, and a covariance of 0 means there is no correlation, that is, there is an independent relationship.
I was talking.
“First, if the GUE hypothesis is correct… It can be assumed that there is a rule in the distribution of prime numbers in a certain interval. At this time, I thought I could find a pattern of covariance values that determine the distribution of these primes. This is a method of weighting using the von Mangoldt function. Let me explain how this weighting is done… … .”
It’s a lot of complicated content, but to put it simply… .
This means that by using the 809 zeros I found, I can explain the statistical distribution among prime numbers in a certain interval.
Of course, since this is a ‘statistic’ distribution, it will be difficult to accurately predict the position of a prime number.
After my explanation, Professor Park thought for a moment before opening his mouth.
“I thought it was sparse… Did you put them together to surprise them all at once?”
“yes?”
“The computer program that draws the zeta function makes my heart pound, but the idea of extending the GUE hypothesis… Heh heh, it seems that Jeong-ho has decided to surprise me today. ha ha ha—–!”
Professor Park laughed for a while.
“I think it will be a really good research topic. It could be a clue to prove the Riemann hypothesis in the future, hmm… So, you mean you want to do this research in the future?”
My answer to the professor’s question is ‘I’ve already done it’.
Using the 809 zeros I found, I have already confirmed the regularity of the distribution of prime numbers.
Of course, since the sample is too small, we need to find more zeros, but it is clear that it is a sufficiently meaningful research result, and it could be published as a single paper.
And the most important conclusion of this paper is… .
It is the fact that the Riemann zeta function expressed in three-dimensional space is connected to three-dimensional or higher dimensions.
This conclusion is somewhat consistent with the string theory that the universe, which appears to be three-dimensional, is actually ten-dimensional.
When asked if I would like to do this research in the future, I said.
“no.”
I have no intention of appearing in the world yet.
so far it is
It will not be too late to show the full-scale research results as an adult.
read at noblemtl.com
Not only this.
In order to properly establish the M-theory, an updated version of the string theory, we have to wait a long time anyway.
I was talking.
“We have only one computer in our house. In fact, even 800 zeros were found by running the computer all week. You’ll probably need more than 10 computers to do a decent statistical study.”
“… … .”
“So then… Can I entrust this research to my brother or sister who is interested in your lab?”
Professor Park said with a disapproving expression.
“Is that okay? This is enough… It’s like you’ve finished preparing the table, isn’t it?”
I shook my head.
“It’s not like that. Did I just give you some ideas? You know the saying, Ideas are cheap!”
Ideas are cheap!
This is a saying a lot among researchers.
This means that the process of verifying the idea and drawing meaningful conclusions is more important than the idea that is the starting point of research.
Professor Park nodded at my words and said.
“I have a master’s student who just started. My name is Yunseok Jo, and I am a student who is very interested in statistics and cryptography, so let’s talk to Yunseok.”
The word cryptography reminded me of a newspaper column I read a while ago.
“I saw the newspaper column written by the professor.”
“ah… ! you saw it too haha!”
The newspaper column was about the RSA cryptosystem.
The RSA cryptosystem is based on a prime number, a very, very large number… For example, use these numbers.
292439209840397098427502985982347672310890456093278586474983458034982307520958679138571093761994319764307561270787642683591
Even a good computer takes a really long time to factor such a huge number, and the RSA encryption system was created based on this characteristic.
For example, the RSA cipher uses a number that is the product of two huge primes, and these two primes are used as the key to the encryption.
So, if you can find these two prime numbers, that is, if the prime factorization succeeds, the password is solved.
There is a myth that the RSA cryptosystem will collapse if the Riemann hypothesis is proven.
I was curious about the opinion of Professor Park, who is majoring in Riemann hypothesis and cryptography, on this ghost story.
For information on extending the GUE hypothesis, refer to the abstract of the paper below.
Brad Rodgers (2013) The statistics of the zeros of the Riemann zeta-function and related topics. PhD thesis, UCLA.