No. Parent has it essentially correct. Many new results in number theory are obtained by studying automorphic forms, which are the stable waveforms, on various spaces.
Things like the Riemann zeta function arise out of spectral transforms of automorphic forms.
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.
The visualization in the OP shows how to use sine waves to build a sieve of Sieve of Eratosthenes. Now that I've seen the visualization, this revelation seems so utterly obvious that it goes without saying. But somehow, I never drew this connection until seeing the visualization.
And once I see how "obvious" this is, it's suddenly obvious how e and pi might get into everything, because everything that repeats with a specific frequency can be modeled as a wheel rolling along and leaving a mark on every revolution. And what is multiplication, but repeated addition? I.e., a wheel of a certain size rolling down the number line, leaving its mark once per turn. Above a certain age, we tend to stop thinking about multiplication as repeated addition, and so we don't think about how all multiplication is implicitly bringing pi into everything we are doing.
Maybe everything I said above is wrong in some way, since, as I have mentioned, I haven't studied any math past calculus and college algebra, and even that was so long ago, most of it I don't remember. Or maybe what I've said is so obvious to someone who has studied math seriously that they just want to shout, "Duh!" But there must be some way to interpret what I just wrote that doesn't deserve being summarily shot down.
> 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 earth can complex numbers prove stuff about modular arithmetic!?)
> Maybe everything I said above is wrong in some way...
Not really. Your intuition upon seeing this visualization was pretty much right on: studying periodic functions is a way of understanding numbers.
> Or maybe what I've said is so obvious to someone who has studied math seriously that they just want to shout, "Duh!"
Not so much. It took some mighty smart folks to develop some ideas which are perhaps suggested, in hindsight, by this picture. The big one is Fourier series and transforms, which allow you to decompose periodic functions into their constituent sine waves. You can use Fourier analysis to get information about number theory, which was essentially your suggestion. However, that's not at all obvious without seeing this picture. Certainly, my first exposures to Fourier analysis were in the context of signal processing and solving PDEs. I had absolutely no inkling that it may be useful for number theory until actually seeing it. Even if I had seen this picture 7 years ago (when I knew signal processing and PDEs, but not number-theoretic applications), I probably would not have made the connection that you made.
So, I think your intuition was a rather non-obvious idea, and so your comment did not deserve the quick shoot-down. (And even if it were obvious to folks who have studied math, it would still be non-obvious to someone, probably).
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!
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.