A second look at the Frequency-of-Primes

An Approach to the Structural Integrity of Prime Number Systems

Section AT-4 - A Second Look at the { Frequency-of-Primes }

Let us recall that we have set up three ( 3) different versions for the { Frequency-of-Primes } :
- First Version , the { PSI-of-(x) } :
  - the standard version consistent with contemporary mathematics .
- Second Version , the { PSI-sub-alpha-of-(x) } :
  - inclusive of the Super-Prime [ 1 ] in the count .
- Third Version , the { PSI-sub-beta-of-(x) } :
  - for the [ Modulo M Prime Number System ] .
And we have broken down the { PSI-sub-Beta-of-(x) } into 2 components ,
- i.e. :
  - where :
    - and
Let us now take a closer look at the ratio [ R2 ] .

Let us take a quick look at the value of [ R2 ] ,
- whenever the value of [ x ] is a { product-of-primes } .
We then set up the value [ X-BAR ] to identify these specific values of [ x ] ,
- i.e. :
And the value of [ X-BAR ] based on [ Q ] is then :
Consequently , we can now set up the value { R-sub-X-of-(Q) } being :
And its relation with [ R2 ] is then :
We have therefore pegged the value of [ R2 ] to the value of { R-sub-X-of(q) }
- at the specific values of [ x = X-BAR ] .

Let us recall that the value of [ R-sub-X ] is given by :
We then have :
- yielding :
We the take the logarithm on both sides to arrive at this equation :
Let us now bring-in this Expansion Formula for the { logarithm of [ 1 + Delta ] } :
Applying this to the equation immediately above then yield us this final equation :

Let us recall that :
- in the Riemann paper of 1859 titled :
  - On the number of Prime Number less than a Given Quantity
  Riemann did start-out with Euler's observation :
  - where :
    - if for [ p ] we substitute each prime number , and for [ n ] every whole number .
Let us bring-in again the { R-sub-X-of-(Q) } :
and look at its reciprical as follows :
Taking the Logarithm of the Reciprical here then yield us this :
This is indeed very interesting equation :
- that requires further studies .