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

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

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Section X - Setting up 3 more Functions

Summary for the Section :

• In this Section , we shall set up 3 more Functions , for the purpose of tracking Euler here in Part IV and Riemann in Part V .
  • The 3 new functions are then :
    • the function [ EZM-of-(s) ] ,
    • the function [ RVM-of-(s) ] , and
    • the function [ LVM-of-(s) ] .

  And we shall also introduce a 4th function , the function [ EZ-Lamda(M)-of-(s) ] , related to the 1st function above .

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Defining the function [ EZM-of-(s) ] :

• For any value of [ M ] greater than or equal to [ 2 ] , we set up the [ EZM-of-(s) ] function via :
  • 

    

  The function [ EZM-of-(s) ] is then a [ Euler Zeta-type function ] for a particular finite value of [ M ] ;

  • and hence the name [ EZM-of-(s) ] .

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Defining the function [ RVM-of-(s) ] :

• For any value of [ M ] greater than or equal to [ 2 ] , we set up the [ RVM-of-(s) ] function via :
  • 

    

    The values for the [ Q-i ]'s here are as defined in Section IV , with the key property here being , again :

    • 

  The function [ RVM-of-(s) ] is then a [ Roving-Value type function ] for a particular finite value of [ M ] ;

  • and hence the name [ RVM-of-(s) ] .

• We take note , at this point , of this standard relation for a finite { geometric series } :
  • 

    We shall be using this for an expansion of the [ RVM-of-(s) ] function in the next Section .

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Defining the function [ LVM-of-(s) ] :

• For any value of [ M ] greater than or equal to [ 2 ] , we set up the [ LVM-of-(s) ] function via :
  • 

    

  The function [ LVM-of-(s) ] is then a [ Limiting-Value type function ] for a particular finite value of [ M ] ;

  • and hence the name [ LVM-of-(s) ] .

• We take note , at this point , of this standard relation for an infinit { geometric series } :
  • 

    This relation is valid as long as the following condition remains satisfied :

    • 

  As such , the function [ LVM-of-(s) ] is the limiting value for the function [ RVM-of-(s) ] ,

  • as we permit the values of all [ Q-i ]'s go to [ infinity ] , everything-all-at-once .

    And we shall come back to this functional relationship in Part V ,

    • when we look more closely at the 1859 Riemann paper on Prime Numbers .

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The 4th Function [ EZ-Lamda(M)-of-(s) ] :

• For the 4th Function , [ EZ-Lamda(M)-of-(s) ] , we have :
  • 

    

  This is the same as the 1st function , [ EZM-of-(s) ] , but the upper-limit for [ summation range ] is now changed to :

  • [ Lamda-of-(M) ] , being the { L.C.M. of natural numbers from [ 2 ] to [ M ] } .

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• We are now ready to look at the Euler Product Formula , next .

go to the next section : Section XI - The Euler Product Formula via the { Fung Inequality }

go to the last section : Section IX - Singularity and Symmetries in { 2-D / 3-D / 4-D Hyperbola's }

return to the Prime Numbers / L.C.M. / Zeta Functions HomePage

original dated 2009-5-08