Is there anything particularly interesting about this graph?
Sum[n=1..infinity] (1/n)*sin(ithprime(n)*pi*x)
Is there?
Does it converge in some points at infinity? Is it possible to find a convergence domain?
Not that I could get anything out of it numbertheoretically speakig...
>>7727576
What?
Can you explain the spikes?
>>7727353
From the graph, there seems to be regular big jump discontinuities.
In between these jump discontinuities, it is twice globally decreasing, then twice globally increasing, then twice globally decreasing.
Do you have the same behaviour at different scales?
The graph is anti-symmetric about 0. Why did you graph twice the same thing?
etc.
I guess we can say that the numerics are sub-optimal, as [math]\sin( k \pi )=0 [/math] for all naturals and so that graph should be 0 at x=1.
Assuming we had the perfect graph of your function, it would say something about the number of primes p with 'p mod 4 = 1' vs. 'p mod 4 = 3', because at x=1/2 the with have an a sum of +-1/n and if there is an imbalance the thing diverges there.
We have
[math] \sin( p_n \pi x) = p_n \, \pi \, x \cdot \prod_{m=1}^{\infty} \left(1 - p_n \, \frac { x } {m} \right) \cdot \prod_{m=1}^{\infty} \left(1 + p_n \, \frac { x } {m} \right) [/math]
and it makes me think of
[math] \prod_{n=1}^{\infty} \left(1 - p_n^{-s} \right) = \frac {1} { \zeta(s) } [/math]
There is
[math] \sin (nx) = 2 \cos (x) \cdot \sin ((n-1) x) - \sin ((n-2) x) [/math]
but i donno...
Please post more analysis of this graph.
Why are there such jumps at 1/3, 2/5, 3/5, 2/3?
Is there a general pattern for all of the smaller jumps?
at 1 there might be a discontinuity, it is plotted as a vertical line but it makes no sense.
This jump will diverge to infinity as n grows big since sin is odd around Pi, and sin((1+epsilon)*pi*PI(n)) = sin(1*pi*PI(n))*cos(epsilon*pi*PI(n)) + cos(1*pi*PI(n))*sin(epsilon*pi*PI(n)).
this leaves us with cos(1*pi*PI(n))*sin(epsilon*pi*PI(n)), the cosine part is of module 1 and constant in sign since primes after 2 are odd, and the sine part goes to epsilon*pi*PI(n) when epsilon is small, which makes everything the sum of harmonic serie multiplied by a factor of epsilon*pi*PI(n), which is infinite.
I may be incorrect and I beg your pardon for this shitty post
>>7727353
The space of continuous real functions on [a,b] with metric given by the integral of (f-g)^2 for elements f and g is incomplete in the sense that Cauchy sequences do not necessarily converge within the space.
You can consider the function f_n(x) = {-1 for x<-(1/n), nx for -(1/n)≤x≤(1/n) and 1 for x>1/n} on the interval [-1,1]. Clearly {f_n} is cauchy, for d(f_n, f_n')<2/min[n,n']. Use the definition of metric and the discontinuous function s sending x to -1 for negative x and to 1 otherwise to write d(f,s)≤d(f,f_n)+d(f_n,s) for arbitrary f. Continuity of f gives that d(f,s) is nonvanishing but d(f_n,s) converges to zero. Hence d(f,f_n) does not converge to zero for any f.
Try to apply a similar argument to your series, say about x=1.
Can someone explain the "plateaus" between the relatively large jumps: x=(1/3) .. (2/5) and x=(3/5) .. (2/3)
And a general pattern for all of the jumps?
>>7729159
Actually this is to be expected because 1, sin(2pi*nx/(b-a)) and cos(2pi*nx/(b-a)) form an orthogonal basis for this space, your ithprime term just chooses which basis elements to use and the /n term scales them.
>>7727353
That's one funky wave. The plotted results even look musical.
Lots of symmetry and similar sequences in there.
>>7729171
Is this not simplistic to answer?
>>7727353
Looks like just a random, whacky fourier series
How come it's such a beautiful graph? There must be something to it. Just too nice to look at.
>>7729171
Please, can someone explain the plateaus?
I'm sure this would be easy and simplistic for you, /sci/
>>7732944
can you off yourself
how can be such an idiot and yet still be alive
>>7727353
This is probably obvious but if you rotate it 180 degrees about the center there then you'll get a symmetry.
>>7732980
Do you mean it's such a simple graph that I should be able to do the (trivial) analysis myself?
is this a Weiner Process?
>>7733995
I don't think so, no.
Please analyze, /sci/
Isn't this superbly interesting?
>>7727353
I don't think so.
Why would there be?
>>7735362
Shows something pattern-esque out of primes.
>4 days later
>nobody has pointed out that this function is related to riemann hypothesis
>>7736202
Tell me more.
>>7736202
How is it related?
>>7733995
Like peeing on Uranus?
>>7729159
And what if we apply it in a Banach space?
