What is this strange constellation of primes found in the plot of $ f(x) = \sum\limits_{n=1}^{\infty} {1 \over 2^{n-1}} e^ {{2\pi i x \over {n}} } $?

Solution 1:

Let us start with $$ f(K) = \sum_{n\ge 1} s^n\; \sin\frac {2\pi i\; K }n\ ,\ s=\frac 12\ ,\ K \in \mathbb Z\ .$$ (Other values of $s$, with a "small" $|s|$ may also be relevant for similar phenomena. Possibly, other sequences $(a_n)$ with $a_n\searrow 0$ lead to the similar behavior. But to fix ideas, $s=1/2$ in the sequel, and $(1/2)^n\searrow 0$ exponentially.)

Let me print the first approximations of the first few sine values $\exp (2\pi i\; K/n)$, $1\le n\le 9$, for the values $29$ and $30$ of $K$. sage code and results:

sage: for K in [28, 29, 30]:
....:     for n in [1..9]:
....:         print "n=%s exp(2 pi i %s/n) ~ %s" % ( n, K, CC( exp(2*pi*i*K/n) ) )
....:     print
....:     
n=1 exp(2 pi i 28/n) ~ 1.00000000000000
n=2 exp(2 pi i 28/n) ~ 1.00000000000000
n=3 exp(2 pi i 28/n) ~ -0.500000000000000 + 0.866025403784439*I
n=4 exp(2 pi i 28/n) ~ 1.00000000000000
n=5 exp(2 pi i 28/n) ~ -0.809016994374947 - 0.587785252292473*I
n=6 exp(2 pi i 28/n) ~ -0.500000000000000 - 0.866025403784439*I
n=7 exp(2 pi i 28/n) ~ 1.00000000000000
n=8 exp(2 pi i 28/n) ~ -1.00000000000000
n=9 exp(2 pi i 28/n) ~ 0.766044443118979 + 0.642787609686538*I

n=1 exp(2 pi i 29/n) ~ 1.00000000000000
n=2 exp(2 pi i 29/n) ~ -1.00000000000000
n=3 exp(2 pi i 29/n) ~ -0.500000000000000 - 0.866025403784439*I
n=4 exp(2 pi i 29/n) ~ 1.00000000000000*I
n=5 exp(2 pi i 29/n) ~ 0.309016994374947 - 0.951056516295154*I
n=6 exp(2 pi i 29/n) ~ 0.500000000000000 - 0.866025403784439*I
n=7 exp(2 pi i 29/n) ~ 0.623489801858732 + 0.781831482468031*I
n=8 exp(2 pi i 29/n) ~ -0.707106781186547 - 0.707106781186547*I
n=9 exp(2 pi i 29/n) ~ 0.173648177666929 + 0.984807753012208*I

n=1 exp(2 pi i 30/n) ~ 1.00000000000000
n=2 exp(2 pi i 30/n) ~ 1.00000000000000
n=3 exp(2 pi i 30/n) ~ 1.00000000000000
n=4 exp(2 pi i 30/n) ~ -1.00000000000000
n=5 exp(2 pi i 30/n) ~ 1.00000000000000
n=6 exp(2 pi i 30/n) ~ 1.00000000000000
n=7 exp(2 pi i 30/n) ~ -0.222520933956313 + 0.974927912181824*I
n=8 exp(2 pi i 30/n) ~ -1.00000000000000*I
n=9 exp(2 pi i 30/n) ~ -0.500000000000000 + 0.866025403784439*I

We can write then the sum for $f(K)$ "numerically" $$ f(K) = \sum_{1\le n\le 6} s^n\; \exp\left( 2\pi i\frac {b(K,n)}n\right) + \underbrace{\sum_{7\le n} s^n\; \exp\left( 2\pi i\frac {b(K,n)}n\right)}_{\text{Noise of order } s^7}\ , \ 0\le b(K,n)< n\ .$$

We can then define "clusters" by ignoring the "noise" starting with some given level, say level $7$ as above first. Then we have to consider all sums of the shape $$ \sum_{1\le n\le 6} s^n\; \exp\left( 2\pi i\frac {t(n)}n\right)\ , \ 0\le t(n)< n\ .$$ There are $60$ such possible sums, where $60$ is structurally obtained as the l.c.m of $1,2,3,4,5,6$.

(For $K=60$, the above sage plot delivers $1,1,1,1,1,1,\dots$, compared to $1,1,1,-1,1,1,\dots$ for $K=30$.)

If we want to zoom in, then the sequence of corresponding l.c.m. values is

sage: L = list( set( [ lcm( [1..N] ) for N in [6..20] ] ) ); L.sort(); L
[60, 420, 840, 2520, 27720, 360360, 720720, 12252240, 232792560]

This controls the "zooming in strategy" and the refined cluster formation.

Note: See also the comment of erick-wong above.