101–105 of 105 posts
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Are you claiming that I've gleaned no insight at all as to why Fourier transforms are used in number theory? Or are you claiming that I just didn't acquire this insight from the visualization? If you claim the latter, then just where did I get it from?
Earlier quoted context omitted.
Are you claiming that I've gleaned no insight at all as to why Fourier transforms are used in number theory? Or are you claiming that I just didn't acquire this insight from the visualization? If you claim the latter, then just where did I get it from?
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Did you look at the paper that the visualization cites? It provides both the semicircular version and a sine-wave version. The paper does some mathematical analysis on the sine wave version, and putatively comes up with a way to transform the series of sine waves for a Sieve of Eratosthenes of a finite size into a single function that is close to zero for composites and significantly non-zero for primes.
I assumed on first sight of the visualization that the semicircles were standing in for sine waves, and I was right. I assumed that semicircles were used because they are easier to draw, or because it's easier to see the Sieve of Eratosthenes in it, or that it was necessary to make this adjustment in order to perform well. Or maybe just that it was prettier. But it was clearly alluding to sine waves. Looking at the paper demonstrates that I was correct on this assumption.
By the way, you do know that any periodic function can be expressed as a sum of sine waves, don't you? Even a wave form made out of repeating semicircles. What I didn't know before this is that Fourier transforms are used in number theory, and now I know, thanks to this visualization. And best of all, this visualization let me intuit that fact on my own. I can't imagine a visualization that could provide anything better than that!
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I clicked on the title "Prime Number Patterns" and am a little disappointed. I'm just not seeing any patterns other than the semicircles of increasing integer diameters. Should I stare at it longer?
Umm... yes, you should actually. Those patterns of semicircles aren't random, of course. They correspond directly to the degree of compositeness of the chosen modulus. Compare for n = 60,61,62, for example. The higher the totient value for n, the more circles you see, basically.
But I can't see a broader pattern beyond that.
Now here I can see some patterns! https://en.wikipedia.org/wiki/Ulam_spiral
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Thank you. I don't know what's up with the increasing trend around here for people to imply you are an idiot over some nitpick that seems to reveal only that the nitpicker spent no effort to try understanding what you had to say, and would rather berate you for a detail rather than engaging in the gist. One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is…
You're welcome! > One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is even just simple things, like how pi and e seem to get into everything, even where you might not naively expect it. Yep. Sure blew my mind when I saw a proof of quadratic reciprocity (a very neat result about square numbers in modular arithmetic) which used complex analysis (how on ear…
This leads me to a question: Do mathematicians actually try to analyze primes by looking at a function that is created by combining a set of sine waves where there is one sine wave for each integer, on order to form a sieve out of the sine waves? E.g., creating a function that crosses zero only at each composite, or some such? Or is this visualization only suggestive of a broad approach?
The paper cited by the visualization is clearly attempting to do what I just described, but it appears to be the work of an amateur, and I don't read Spanish, so I can't really tell if this approach is on sound footing. I tried to Google around looking for this approach referenced in something more authoritative, but couldn't find any. I did find plenty of references to trying to analyze the function that you get from subtracting x/ln(x) from the prime staircase, using Fourier transforms and the like. But I can't see a direct connection between these approaches, other than the general inspiration of trying to break the problem down into a combination of sine waves. On the other hand, I'm well aware that a lot of identities in math are not readily obvious!
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You're welcome! > One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is even just simple things, like how pi and e seem to get into everything, even where you might not naively expect it. Yep. Sure blew my mind when I saw a proof of quadratic reciprocity (a very neat result about square numbers in modular arithmetic) which used complex analysis (how on ear…
Thanks again! You restore my faith that it is possible to have a reasonable conversation around here. This leads me to a question: Do mathematicians actually try to analyze primes by looking at a function that is created by combining a set of sine waves where there is one sine wave for each integer, on order to form a sieve out of the sine waves? E.g., creating a function that crosses zero only at each composite, or…
The stable waves on a circle are precisely those waves which oscillate an integer number of times as they traverse. In other words, one for each integer. Further, every function on the circle can be expressed as a sum of the sine waves. That sum is called the spectral decomposition. (This is Fourier series.)
With clever choices of functions, you can get some profound results. For example, picking a saw-tooth wave and doing the spectral decomposition gives the identity
1 + 1/4 + 1/9 + 1/16 + ... = pi^2/6
And, by the way, the left hand side is the zeta function evaluated at 2.
And about functions that are zero at each composite... You may want to check out Dirichlet characters. They are periodic functions which behave nicely under multiplication. Whenever an integer and the period have a common factor, the character will be zero at that integer.
It's not going to be zero at all composites, but it's on the right track.