An Approach to the Structural Integrity of Prime Number Systems

by Frank C. Fung ( 1st published in October, 2012. )

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Section AT-1 - The { Frequency of Prime Numbers } in general

Summary for the Section :

• In this Section , we shall set up the definition for the { Frequency-o-Primes } ,
  • following closely the [ norm ] of contemporary mathematics .

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The General Definition for the [ Frequecy of Prime Numbers ] :

• We bring-in , first-of-all , the function [ pi-of-(x) ] :
  • 

    the counting function for Prime Numbers up to a given value [ x ] .

• We then define the { Frequency-of-Primes up to [ x ] } as :
  • 

    yielding :

  • 

  We shall then label this function as the function { PSI-of-(x) } :

  • 

    for identification in our analysis to follow .

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Using an Adjusted Version for the { Frequency-Of-Primes } :

• Let us recall at this point that :
  • { natural numbers up to [ x ] } are always made up of these three (3) components :
    • the number [ 1 ] as the Super-Prime ,
    • [ prime numbers ] up to [ x ] , and
    • [ composite numebrs ] up to [ x ] .

  And for our analysis to follow ,

  • we do propose to treat the Super-Prime [ 1 ] as a quasi-prime-number ,
    • and include it into our count of [ prime-numbers and quasi-prime-numbers ] ,

    to come up with this adjusted version for the { PSI-of-(x) } , as follows :

    • 

  And this adjusted version for the { Frequency-fo-Primes } shall from-here-on-in be knowns as :

  • the { PSI-sub-alpha-of-(x) } .

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go to the next Section : Section AT-2 - The { Frequency-of-Primes } for [ Modulo M ]

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Original dated 2012-10-23