What does the Riemann zeta function have to do with the distribution of primes?(hidden-phenomena.com) |
What does the Riemann zeta function have to do with the distribution of primes?(hidden-phenomena.com) |
The article itself feels a bit long without a satisfying payoff at end. But then again, the journey is entertaining even for a non-specialist, and the lack of a strong conclusion is due to the unsolved problem of the Riemann hypothesis. The open-ended question is probably irresistible for some personality types that can't stand the suspense and demand a resolution.
Perhaps a way to strengthen the ending is to explain why the question of the distribution of prime numbers is worth solving, and what are the larger implications.
I skimmed the article, but the next third of the article seems to be devoted to using this relationship between the zeta function and prime numbers to prove the prime number theorem, which is a theorem approximating how many primes are less than or equal to any given number N.
The final third goes into how to get increasingly accurate approximations for the number of primes less than N, ending on the fact that Gausses approximation is in some sense the “best”, but only if the Riemann zeta functions zeroes lie on the critical section.
If you just want a general primer on why the zeta function has anything to do with primes, the product formula might suffice. In which case, the proof on the Wikipedia page might be a better read. The derivation in the article focuses on the general setup that is later built on to prove additional things
Highly simplified, and also wrong, you don't get the actual primes, but an error correcting term to another prime estimation function.
So in a way, the zeroes of the zeta function encode "the frequencies of the primes" (in the Fourier sense)
[1] https://en.wikipedia.org/wiki/Von_Mangoldt_function
[2] Prime numbers and the Riemann hypothesis. Mazur, Stein
> This equation might seem a little hard to solve, but at this point you might notice something funny: $equation$
> So, if F′(x)log(x)=1,F′(x)log(x)=1, then
> Thus, our mystery function F(x)F(x) obeys F′(x)log(x)=1.F′(x)log(x)=1. From here you deduce F′(x)=1/log(x),F′(x)=1/log(x), so by integrating you get
But it does not seems that F needs to have that trait, just that 1 works in that instance. It's sufficient but not necessary. How can you tell that it is that simple solution which is the right one?
Easy to follow without requiring advanced math, great visualizations.
https://www.youtube.com/@ZetaExplained/videos
However needing tens of hours of video to explain what the Riemann Hypothesis is, without glossing over details, tells you something about it's difficulty (as a statement).
5 4 3
6 1 2
7 8 9
> According to Martin Gardner, Ulam discovered the spiral in 1963 while doodling during the presentation of "a long and very boring paper" at a scientific meeting. These hand calculations amounted to "a few hundred points". Shortly afterwards, Ulam and collaborators used MANIAC II at Los Alamos Scientific Laboratory to extend the calculation to about 100,000 points.
Zeta(z) = 1 + 1/2^z + 1/3^z + …
Where z in C.
In fact, I thought that was why it’s called the Riemann zeta function. Euler applied it to an integer whereas Riemann applied it to complex arguments.
Edit to add: my memory was correct. Reimann extended Euler’s definition to all complex s not equal to 1. https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_T...
This didn't seem especially controversial of a comment, and even here there's no end of the rotten attitudes for no good reason
Read-only or read never is looking wiser than it did a few years ago.
Google summarizes the controversy thus, from Wikipedia:
"John Derbyshire is an American journalist and political commentator. He was one of the last paleoconservatives at the National Review, until he was fired in 2012 for writing an article for Taki's Magazine that was widely described as racist. Since 2012 he has written for white nationalist website VDARE. Wikipedia
JESERAC SAT MOTIONLESS within a whirlpool of numbers. The first thousand primes, expressed in the binary scale that had been used for all aritmetical operations since electronic computers were invented, marched in order before him. Endless ranks of 1's and 0's paraded past, bringing before Jeserac's eyes the complete sequence of all those numbers that possessed no factors except themselves and unity. There was a mystery about the primes that had always fascinated Man, and they held his imagination still.
Jeserac was no mathematician, though sometimes he liked to believe he was. All he could do was to search among the infinite array of primes for special relationships and rules which more talented men might incorporate in general laws. He could find how numbers behaved, but he could not explain why. It was his pleasure to hack his way through the arithmetical jungle, and sometimes he discovered wonders that more skillful explorers had missed.
He set up the matrix of all possible integers, and started his computer stringing the primes across its surface as beads might be arranged at the intersections of a mesh. Jeserac had done this a hundred times before, and it had never taught him anything. But he was fascinated by the way in which the numbers he was studying were scattered, apparently according to no laws, across the spectrum of the integers. He knew the laws of distribution that had already been discovered, but always hoped to discover more.
He could scarcely complain about the interruption. If he had wished to remain undisturbed, he should have set his annunciator accordingly. As the gentle chime sounded in his ear, the wall of numbers shivered, the digits blurred together, and Jeserac returned to the world of mere reality.