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 III - Setting up the L.C.M.-based Function [ Lamda-of-(M) ]

Summary for the Section :

• In this Section , we shall set up and formally define the L.C.M.-based Function [ Lamda-of-(M) ] :
  • 

  This function is simply core-and-central to our investigation of { prime numbers } .

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Setting up the Lamda Function :

• For any { positive integer } [ M ] greater than or equal to [ 2 ] , i.e. :
  • 

  we shall define the function [ Lamda-of-(M) ] as follows :

  • 

    

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A few Quick Examples here :

• Let us now cite several specifc values for the [ Lamda-of-(M) ] , as follows :
  • 

  • 

  • 

  • 

    as a first illustration of the function .

• And we take note , at this point , that the first 14 { prime numbers } are , as always :
  • 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 and 43 ;

    so that , in our calculation process for the 4 values above , we can write :

  • 

  • 

  • 

  • 

• We now make these 3 rather interesting observations , namely :
  • 

  • 

  • 

  A very brief explanation is in-order here :

  • For the numbers from [ 33 ] to [ 36 ] :
    • [ 33 ] = 3 x 11
    • [ 34 ] = 2 x 17
    • [ 35 ] = 5 x 7
    • [ 36 ] = 4 x 9

      so that these four (4) numbers are already covered within the L.C.M. [ Lamda-of-(32) ] .

    And as such , the value for the [ Lamda-of-(M) ] function remains constant / unchanged for the numbers [ 32 ] thru [ 36 ] .

  • But the next number [ 37 ] is a { prime number } and this is not covered by the L.C.M. [ Lamda-of-(32) ] .
    • And as such , the value for the [ Lamda-of-(M) ] function must change ( increase ) at this prime number [ 37 ] .

  • For the numbers from [ 38 ] to [ 40 ] :
    • [ 38 ] = 2 x 19
    • [ 39 ] = 3 x 13
    • [ 40 ] = 8 x 5

      so that these three (3) numbers are already covered within the L.C.M. [ Lamda-of-(37) ] .

    And as such , the value for the [ Lamda-of-(M) ] function remains constant / unchanged for the numbers [ 37 ] thru [ 40 ] .

  • But the next number [ 41 ] is a { prime number } and this is not covered by the L.C.M. [ Lamda-of-(37) ] .
    • And as such , the value for the [ Lamda-of-(M) ] function must change ( increase ) at this prime number [ 41 ] .

• This then bring us to the first-and-most-important property for the Lamda Function , next .

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The First-And-Most-Important Property :

• The function [ Lamda-of-(M) ] is in fact { step-function } :
  • the value for [ Lamda-of-(M) ] increases / jumps :
    • whenever the value of [ M ] hits a { prime number } or alternately a { power-of-prime } ;

  • otherwise the value of [ Lamda-of-(M) ] remains the same ,
    • i.e. : unchanged / retaining the same value for function at [ M minus 1 ] .

• But we shall provide more details on this in the next Section .

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go to the next section : Section IV - Setting up the [ QTT-i ]'s for any given value of [ M ]

go to the last section : Section II - A Special Note on the Notations in this paper

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

original dated 2009-5-08