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March 3rd, 2019, 08:07 AM   #1
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Wave Divisor Function

Dear Math Forum,

For quite some time (years), I have been playing around with the divisor function (counting the number of divisors for a given integer). I created a 10 slide summary in the attached presentation.

Wave Divisor Function:
https://drive.google.com/open?id=1lByldG7EWxDsouec1EXBM2yPS88En0RG


I have been able to describe the divisor function with the help of waves (periodic functions). This wave description (now Re and Im) introduces an error in the (Re) solution. However, this error seems proportional to the mean divisor count.

With the described method, it might (maybe) be possible to numerically refine/determine the non-leading terms of the Divisor Summation

Every time I am working on this subject, I have the feeling this is an original way to look at the divisor function (as waves). I make up the idea that the discrete math can be expressed as waves and vice versa (analogue as quantum mechanics).

My wish and hope a mathematician has a look on the attached summary of my findings.

(My skills are too limited to continue any further.)

Vince

Last edited by skipjack; March 3rd, 2019 at 01:21 PM.
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March 10th, 2019, 09:02 AM   #2
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Hello,

My conclusion of the wave divisor function:

A method is described to express the divisor function as a summation of waves function. This wave formulation results in a real and imaginary divisor solution. The divisor count in the wave representation will have and error. The variance of the error is estimated to be proportional to the mean divisor count.

The variance of the error in the wave divisor model can be determined for unlimited pulse width settings. If the error in the divisor model is truly proportional to the mean divisor count this method might be useful refining the non leading terms of the mean divisor function.


Though, my math skills are to limited to come to an prove.

I was not happy how I justified the: n choose k is similar with the trigonometric notation. I adapted the document on small issues (concept / conclusion remains the same):

https://drive.google.com/open?id=1uS...PfVYnsKhe1TU-w

Greetings,

Vince
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