Error symmetry wave divisor function.
The divisor function can be written as a summation of waves (see link below previous questions Stacks Exchange). The error in the wave divisor function is mainly determined by it's neighbor divisors. The error is proportional to:
$$\varepsilon (x) \propto \sum_{\mathbb{X}\vert (x-1)}^{} \cos(k \mathbb{X}) + \sum_{\mathbb{X}\vert (x+1)}^{} \cos(k \mathbb{X})$$
Here $\mathbb{X} \vert (x-1)$ means: $\mathbb{X}$ divides $(x-1)$. Basically the divisors are added from the neighbors of $x$:
$$\varepsilon (9) = \cos(k1)+\cos(k2)+\cos(k4)+\cos(k8)+\cos(k1)+\cos(k2)+\cos(k5)+\cos(k10)$$
The total error then is a (cosine)summation of errors like Brownian motion. Where k is a constant and determines the pulse width of each divisor wave, see link below for more information.
$$k=-\frac{2 \log(L)}{\pi \Delta x^{2}}$$
We can simulate the error for a number $x$ by keeping $L=0.5$ and vary $\Delta x$ between: 0.15 and 0.2 in 10000 steps. For every $k$ the error can be calculated. See simulation below.
It is observed that for $x=odd$ the error $\varepsilon (x)$ tends to nonsymmetrical/skewed distribution. For $x=even$ the error $\varepsilon (x)$ tends to a symmetrical distribution.
Normally I exclude 1 as an divisor, but the symmetrical and skewed distribution are always present (with and without 1 as divisor).
Typical example Symmetrical even $x$:
Typical Example Non Symmetrical odd $x$:
Question:
Why does the error for odd and even numbers $x$ tend to behave symmetric and asymmetric/skewed?
More information.
More information stackexchange: Error in Divisor Function Modelled With Waves
Interactive Simulation Github/Mybinder: Mybinder Jupyternotebook
Partial Answer Skew Error Distribution.
This is the divisor counting of neighbors left and right of x. So sum of divisors at (x-1) and (x+1).
Assumption 1:
Divisors of odd numbers will always be odd: Do odd numbers have only odd divisors?
Case 1 (odd divisors only):
Function analysis show that the following function is symmetrical (positive and negative spikes occur). $$\varepsilon(k) = \sum_{\mathbb{X}}^{30} \cos(k (2\mathbb{X}-1))$$
Case 2 (even divisors only):
Function analysis show that the following function only has positive spikes. $$\varepsilon(k) = \sum_{\mathbb{X}}^{30} \cos(k 2 \mathbb{X})$$
So for even divisors the distribution will look skewed asymmetrical. Even divisors have a mix of odd and even divisors.
Thus the error for even numbers $x$ is symmetric and the error for odd numbers $x$ is skewed.
Though no proof is supplied for both formula above. Maybe the derivative (and determine max and min) supply the proof.