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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 $x$:

enter image description here

Typical Example Non Symmetrical odd $x$:

enter image description here

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

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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.

enter image description here

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