On Unique-Factorization Combinatorics via the Pascal Triangle

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

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Section XXII - Expressing [ z ] as a function of [ H ] and [ T ]

Summary for the Section :

• In this Section , we shall take a look at 2 issues arising from :
  • expressing [ z ] as a function of [ H ] and [ T ] .

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Setting up the general equation :

• Let us recall that we do have a formula now to calculate :
  • the Full Count (z) of { Catgegory [ T ] Composite Numbers } that can be formulated
    • with an initial set of [ H ] distinct-and-different prime-numbers .

  And we can us this general format for the equation :

  • 

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The Issue on Matrix Notations :

• Let us bring-in again the set of 2 diagrams :
  • with the [ ALPHA Matrix ] on-the-left and the [ ETA Matrix ] on-the-right .

  

  Let us recall at this point that :

  • each of the 10 rows of the [ ALPHA Matrix ] is a vector in a [ 10-Dimensionl Space ] ,

    and

  • each of the 10 columns of the [ ETA Matrix ] is also vector in a [ 10-Dimensionl Space ] ,

  And since the [ Z Matrix ] is the product of the [ ALPHA Matrix ] and the [ ETA Matrix ] ,

  • the [ Z Matrix ] is in fact the value of the [ DOT Product ] of 2 Vectors ,
    • one from the [ ALPHA Matrix ] and the other from the [ ETA Matrix ] .

• And the thought being given here is that :
  • IF :
    • We join the 10 end-points of the 10 vectors in the [ ALPHA Matrix ] with a smooth curve ,

      and

      we also join the 10 end-points of the 10 vectors in the [ ETA Matrix ] with a smooth curve ;

    THEN :

    • we might be able to express the relation :
      • 

        with a smooth surface in a { 3-Dimensional Space } .

  But of course , we will always face the problems of :

  • deciding on the smooth curves we might wish to use ,

    and

  • calibrating the intermediate values in-bewteen the [ integer-values ] for both [ H ] and [ T ] .

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The Issue of overall Continuity :

• Let us recall that the Factorial Function is set up via these core governing relations :
  • 

    and

  • 

  And in the Riemann paper of 1859 titled :

  • [ On the Number of Prime Numbers less than a Given Quantity ] ,

    Riemann did bring-in the { PI Function } :

    • 

  And of course the [ PI Function ] co-incides with the [ Factorial Fucntion ]

  • whenever the value of [ s ] is a [ positive integer ] ,

    i.e. :

    • 

• Just wondering as to whether we should do the same for :
  • 

  But the Author do not have to decide on this one

  • until we launch a full attack on the Riemann Zeta Function at a later stage .

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go to the next section : Section XXIII - Concluding Remarks

go to the last section : Section XXI - Symmetry Considerations for the [ Z Matrix ]

return to the Unique-Factorization Combinatorics HomePage

Original dated 2012-10-08