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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 t

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 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!

yaks_hairbrush 14y ago
Oooh... you're getting into some serious math now. 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.

yaks_hairbrush 14y ago

Oooh... you're getting into some serious math now. 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.