SRL Archive - NOTA

Hello,

I shall propose a method for dealing with prime functions.

First off, you must understand what is represented by a characteristic function.

Given a subset A of a larger set, a characteristic function X(a) is defined to be identically one on A, and is zero elsewhere.

Please refer to this source for further information. (Wolfram)

Now consider the set [1, 2, 3,…,20] for f(n) {1 = prime; 0 = not prime}

The resulting values will be = [0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1]

The binary fraction of these values is approximately = 2/5. However, as n approaches infinity, the binary fraction becomes approximately .41468250985111166(…) I will now refer to this value as Cp.

For every prime less than one million (78,498 primes), the value of n was plotted against 1/Prime(n).

{ Prime(n) = the nth prime }

An extremely strong, negative correlation appears between the values of 1/prime(n) and n. The correlation coefficient for the equation is .9999

The currently calculated equation for the inverse of the nth prime is = .2616*n^(-1.108).

I suggest that the integral of this equation from a lower limit of 1 to an upper limit of infinity will be equivalent to 1/Cp.


Or with a more broad scope, the integral of the inverse of a function to calculate the nth prime from [1, infinity] will be equal to 1/Cp.


Comments?

Interesting. When would you ever have to use this?

From: The Mayor
Interesting. When would you ever have to use this?

This is purely theoretical mathematics, but it has potential applications in prime counting functions.