Wave Divisor Function

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Dec 2014
51
12
Netherlands
Error Divisor Summatory Function

Hello,

Did allot of calculations last days. Did not see any patterns and wanted to throw all in bin.

I think I found a way to numerical determine error in Divisor Summatory function (Dirichlet).

$\Theta(x \rightarrow 0) \approx 0.5$

$\Theta_{min}(x \approx 11) \approx 0.26$

Divicor Summatory Theta.jpg

For calculations see Jupyter notebook:

https://mybinder.org/v2/gh/oooVincentooo/Shared/master?filepath=Wave%20Divisor%20Function%20rev%202.4.ipynb

My holiday's is almost over, last weekend. Hoped that someone could verify or look into results.

Best regards,

Vincent Preemen
 
Dec 2014
51
12
Netherlands
Results are wrong made some mistakes.

Was just luck achieving these results.

Do understand even less.

Need more time to think. Time out for now! :rolleyes:
 
Dec 2014
51
12
Netherlands
Wave Divisor Function Arcsine Distribution

Hello,

Determining \(\displaystyle \Theta\) from divisor summatory requires more thinking of me. There are many solutions for the divisor function. My simple "math" skills reaches it's limit.

In the meantime I studied on some of the properties of the arcsine distribution. To my understanding for $\infty$ the error is arcsine distributed.

There are some interesting properties. Demonstrated in the Jupyter notebook two cases of the acrsine distribution:

- Without resonation and with resonation.

As screenshot:
Wave Divisor Function Arcsine.jpg


Jupyter Notebook (Arcsine):
https://mybinder.org/v2/gh/oooVincentooo/Shared/master?filepath=Wave%20Divisor%20Function%20Arcsine.ipynb

Hope that someone has some input or advice.

Jupyter notebooks should open online (in cloud) and are interactive after loading Menu: [Cell]+>[Run All]. Does not work with IE, other browser do. First loading might take some time.

Best regards,

Vincent

Note:
I also updated main notebook file. Fixed some typo's and improved description of the error:
https://mybinder.org/v2/gh/oooVincentooo/Shared/master?filepath=Wave%20Divisor%20Function%20rev%202.4.ipynb
 
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greg1313

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