An Approach to Prime Numbers , the L.C.M. , and the Zeta Functions

by Frank C. Fung - 1st published in May, 2009.

This paper is dedicated to my father , the late Charles Hok-Ling Fung ( 1909-2002 ) , on the occassion of his Centenial Year .

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Summary & Key Findings

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Summary and Key Findings :

• Finding #1 - the L.C.M. as an indicator :

  • Take the 30 consecutive { natural numbers } from [ 2 ] to [ 31 ] :
    • If all 30 numbers are { prime numbers } , the L.C.M. would have been [ 31-Factorial ] ;

    • If the { prime factors } of the 30 numbers comes-down to one single { prime number } , the L.C.M. would have been [ 31 ] ;
      • [ 31 ] being the largest number of the lot .

    The fact the L.C.M. is somewhere in-between serves as a first-indicator as to how many { prime numbers } there are .

• Finding #2 - the lower-bound estimate of [ pi-of-(M) ] via the L.C.M. :

  • We set up the L.C.M.-based function { Lamda-of-(M) } :
    • 

    And we were able to establish :

    • 

      where [ pi-of-(M) ] is the { prime number counting function } .

    The term on-the-very-right under the [ product sign ] is then the [ product-of-primes ] up to [ M ] .

• Finding #3 - A Fast-Track Approach to identifying { prime numbers } :

  • [ 2 x 3 ] = 6 , so that [ 1 mod 6 ] and [ 5 mod 6 ] are candidates for { prime numbers } next .

    We then use the { product-of-primes } series as the [ modulo's ] series to further identify { prime numbers } , i.e. the series :

    • [ modulo 6 / 30 / 210 / 2310 / 30030 / 510510 / 9699690 / etc. ] ;

      with some success initially .

• Finding #4 - Proposing an Inequality in lieu of the Euler Product Formula :

  • In lieu of the Euler Product Formula for { real values of [ s ] } :
    • 

    we propose the { Fung Inequality } , as follows :

    • 

• Finding #5 - Step Functions that pops at { primes } and { power-of-primes } :

  • Riemann's {prime counting function } is the [ F(x) ] function and he used its derived function [ f(x) ] :
    • 

      as the core function in his investigations of { prime numbers } .

    The f(x) fucntion is a { step function } that [ pops ] whenever the value of [ x ] hits a { prime } or a { power-of-prime } .

    • So does our [ Lamda-of-(M) ] function .

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Other Sections of the paper :

Table Of Content

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go to the next section : Section I - Introduction and { a-priori assumptions }

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original dated 2009-5-08 *** Updated 2009-5-17 / 6-02 / 10-26