Stupid question on toilet: if prime numbers are 1s and other integers are 0s. What do we get from this “digital” landscape?
The decimal expansion of an irrational number https://oeis.org/A010051
What does the Riemann zeta function have to do with the distribution of primes?
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Re: What does the Riemann zeta function have to do with the distribution of primes?
#12Stupid question on toilet: if prime numbers are 1s and other integers are 0s. What do we get from this “digital” landscape?
If you plot the positive integers two-dimensionally in a square spiral arrangement, eg. 5 4 3 6 1 2 7 8 9 and so on, and mark all primes, what you get is the Ulam spiral: https://en.wikipedia.org/wiki/Ulam_spiral
I was curious why/how someone would even think about arranging numbers in a spiral. The origin story is funny:
> 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.
Re: What does the Riemann zeta function have to do with the distribution of primes?
#13Although the author turned out to be a fairly despicable person, Prime Obsession is an absolutely wonderful book on this subject.
I despise vague and quasi-anonymous attacks like this. "Trust me, online person, I, a random netizen, have made a moral judgment and you should have complete faith in it."
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
Re: What does the Riemann zeta function have to do with the distribution of primes?
#14And as if all of that were not head-exploding enough, I'm still searching for an implementation of the zeta function for complex arguments...
It doesn’t change if you apply it to complex arguments does it?
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...
Re: What does the Riemann zeta function have to do with the distribution of primes?
#15I’ve seen the connection between the Riemann zeta function and primes talked about but never an accessible explanation. Although I must say that the beginning was fairly easy to follow, I got overwhelmed about a quarter into it. Feels like I need a week of study to really comprehend the entire article.
Re: What does the Riemann zeta function have to do with the distribution of primes?
#16Although the author turned out to be a fairly despicable person, Prime Obsession is an absolutely wonderful book on this subject.
I despise vague and quasi-anonymous attacks like this. "Trust me, online person, I, a random netizen, have made a moral judgment and you should have complete faith in it." 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 Ta…
Would you prefer OC was more specific, ie “the author turned out to be a racist”, or do you have an issue with calling someone like this despicable?
Re: What does the Riemann zeta function have to do with the distribution of primes?
#17I’ve seen the connection between the Riemann zeta function and primes talked about but never an accessible explanation. Although I must say that the beginning was fairly easy to follow, I got overwhelmed about a quarter into it. Feels like I need a week of study to really comprehend the entire article.
Had same problem. One thing which is not always made clear: sound waves can be thought as a sum of sinusoidals of different frequencies (the Fourier Transform). The zeta functions does something similar: you add up the zeta functions for the non-trivial zeros, and the sum of them has jumps at the location of primes.
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)
Re: What does the Riemann zeta function have to do with the distribution of primes?
#18There is a whole YouTube channel just about this subject. Tens of hours in total, in 30-50 min episodes handling little chunks of this matter (Zeta/Riemann/primes)
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).
Re: What does the Riemann zeta function have to do with the distribution of primes?
#19Personally I’ve found this article a much more approachable overview to the question in the title: https://golem.ph.utexas.edu/category/2019/09/the_riemann_hyp...
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Re: What does the Riemann zeta function have to do with the distribution of primes?
#20There is a whole YouTube channel just about this subject. Tens of hours in total, in 30-50 min episodes handling little chunks of this matter (Zeta/Riemann/primes) 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 i…
I don't think tens of hours are necessary. There was a very good public talk by a mathematician from first principles that even a child can understand, all the way up to the Riemann hypothesis, in 50 minutes. I wish it was online. It was the best talk of any subject I have ever heard.