Divisor Function Symmetry Neighbor Divisors

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Dec 2014
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Divisor Function Symmetry Neighbor Divisors.

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 neigbor divisors. The error is proportional to:

\(\displaystyle \varepsilon (x) \propto \sum_{\mathbb{X}\vert (x-1)}^{} \cos(k \mathbb{X}) + \sum_{\mathbb{X}\vert (x+1)}^{} \cos(k \mathbb{X})\)

Here $\mathbb{X} \vert (x-1)$ means: $\mathbb{X}$ divides $(x-1)$. Basically the divisors are added from the neighbors of $x$. The total error is basically a 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.

\(\displaystyle k=-\frac{2 \log(L)}{\pi \Delta x^{2}}\)

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. See simulation below. It is observed that for $x=odd$ the error $\varepsilon (x)$ tends to nonsymmetrical distribution. For $x=even$ the error $\varepsilon (x)$ tends to a symmetrical distribution. My excuses if the math notation might be non-standard. This is a hobby project of mine.

Typical example Symmetrical even $x$:

Even x.png

Typical example Non Symmetrical odd $x$:

Odd x.png

Question:
Why does the error for odd and even numbers $x$ tend to behave symmetric and asymmetric?

Sorry just cannot leave the topic alone. I have holiday and hope to learn more about the wave divisor function. Also posted question on Exchange but don't expect an answer. Hope that someone here has some clues about symmetry and skew. I attempted to phrase this post so no previous information is required.

More links and Jupyter notebook simulations can be found on stackexchange. More information:
Divisor Function Symmetry Neighbor Divisors
 
Dec 2014
51
12
Netherlands
I think I explained the problem to complex. I rephrased a little with an example:

\(\displaystyle \varepsilon (x) \propto \sum_{\mathbb{X}\vert (x-1)}^{} \cos(k \mathbb{X}) + \sum_{\mathbb{X}\vert (x+1)}^{} \cos(k \mathbb{X})\)

Here $\mathbb{X} \vert (x-1)$ means: $\mathbb{X}$ divides $(x-1)$. Basically the divisors are added from the neighbors of $x$:

\(\displaystyle \varepsilon (9) = \cos(k1)+\cos(k2)+\cos(k4)+\cos(k8)+\cos(k1)+\cos(k2)+\cos(k5)+\cos(k10)\)

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, more information/link first post.

\(\displaystyle k=-\frac{2 \log(L)}{\pi \Delta x^{2}}\)

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. Rest see above (first post).

Normally I exclude 1 as an divisor, but the symmetrical and skewed distribution are always present (with and without 1 as divisor).

The main question is:
Why does the distribution for odd $x$ tends to asymmetry/skew and for even $x$ symmetrical.
 
Dec 2014
51
12
Netherlands
btw. This is the divisor counting of neighbors left and right of x. So sum of divisors at (x-1) and (x+1). A odd x always has p=2 as neighbor left and right. Basically:

\(\displaystyle 2\cos(k2)\)

Do you think the cosine sum has more positive values when p=2 or multiples?

I was just thinking/browsing and found the following identity:

\(\displaystyle \cos(2\Theta)=1-2\sin^{2}(\Theta)\)

Maybe that for odd x's, that sums the divisors of even (neigbor) numbers. Maybe that the identity above occurs more often for divisors of even numbers? That means the error is more often positive while it's squared?

Not sure though need more thinking :rolleyes:. To complex for my brain.
 
Dec 2014
51
12
Netherlands
Likely 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 number 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).

\(\displaystyle \varepsilon(k) = \sum_{\mathbb{X}_{odd}} \cos(k \mathbb{X}_{odd})\)

***Case 2 (even divisors only):***
Function analysis show that the following function only has positive spikes.

\(\displaystyle \varepsilon(k) = \sum_{\mathbb{X}_{even}} \cos(k \mathbb{X}_{even})\)

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 I do not have a proof for case 1 and case 2. Next on list! :D
 
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Dec 2014
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Here a plot from the odd and even function:

Odd: \(\displaystyle \varepsilon(k) = \sum_{\mathbb{X}}^{30} \cos(k (2\mathbb{X}-1))\)

Even: \(\displaystyle \varepsilon(k) = \sum_{\mathbb{X}}^{30} \cos(k 2 \mathbb{X})\)

Not sure how to proof even is asymmetric in y-axes and odd is symmetric in y-axis. But I am happy I found another clue.

Though no proof is supplied for both formula above. Maybe the derivative (and determine max and min) supply the proof.


Odd Even Cosine.png
 
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greg1313

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Unfortunately, this is not the place for these types of posts. Thread closed.

Please PM me if you have any questions.
 
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