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 XII - Riemann's start-off with the Euler Observation

Summary for the Section :

• In this Section , we shall look closely at start-off position for Riemann's 1859 paper on { prime numbers } .

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The Riemann paper of 1859 :

• In his paper on { Prime Numbers } in 1859 , Riemann started-off with this observation from Euler :
  • 

  where :

  • on the left-hand-side : [ p ] is each a { prime number } ,

  • on the right-hand-side : this involves all { natural numbers } .

• And Riemann said , basically :
  • whenever the two (2) sides converge , he shall call it the { Zeta Function } .

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Concentrating on the Half-Plane :

• Riemann then asserted that the 2 sides do converge :
  • Whenever the { real part of [ s ] } is greater than [ 1 ] ,
    • the left-hand-side and the right-hand-side do converge .

  The remainder of Riemann's paper is primarily concerned with the half-plane :

  • 

  Our proposals below , for both the left-hand-side and the right-hand-side , will follow suit :

  • and will concentrate on the behavior { Zeta Function } on the said half-plane .

Proposal for the Left-Hand-Side :

• For the left-hand-side , we now propose to use :
  • either :
    • the [ RVM-of-(s) ] function , from Section X ;

    or :

    • the [ LVM-of-(s) ] function , also from Section X ;

    before we let the value of [ M ] move-out towards [ infinity ] to arrive at the original left-hand-side above .

• For the [ RVM-of-(s) ] function , we have :
  • 

    And the expansion here is via { geometric series with a finite number of terms } , i.e. :

    • 

• For the [ LVM-of-(s) ] function , we have :
  • 

    And the expansion here is via { geometric series with an infinite number of terms } , i.e. :

    • 

• We take note , at this point , that for the { prime number } [ 2 ] :
  • expansion via either of the 2 series does yield a finite result , as follows :
    • for the finite series involving [ RVM-of-(s) ] , we have this , as an example :
      • 

        

        And the value here is a [ finite value ] , complex or otherwise .

    • for the infinit series involving [ LVM-of-(s) ] , we have :
      • 

        

        Again the value here is a [ finite value ] , complex or otherwise .

  In fact , for any given { finite prime number } [ p-i ] :

  • the expansion via either of the 2 series does yield absolutely a [ finite result ] , on the said half-plane .

• Let us assume , for the moment , that we are looking only at { prime numbers } up to a finite value [ M ] :
  • We are therefore looking at a finite number of { prime numbers } .

  As such , when we use either the [ RVM-of-(s) ] or the [ LVM-of-(s) ] function for the Left-Hand-Side , i.e. :

  • 

    or :

  • 

    we do end up with a [ finite value ] , complex or otherwise , for the entire function ,

    • on the half-plane :

      

      as long the value of [ M ] remains finite .

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The Right-Hand-Side :

• For the proposal on the right-hand-side , let us first bring back the { Fung Inequality } from the last Section , Section XI :
  • 

    

  But this equation also says this :

  • There exists a value [ V ] , such that :
    • 

    And the value [ V ] is in-between [ M ] and [ Lamda-of-(M) ] , i.e. governed by :

    • 

• The simple question then arises :
  • If we were to use this function [ Zeta-V-of-(s) ] below for the right-hand-side :
    • 

    then :

    • is there any possibility that we can find a value of [ V ] , so that :
      • the complex value here is sufficiently close to the left-hand-side ,
        • to warrant us to say that the 2 sides have converged ?

  • A good question , but remains unanswered .

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go to the next section : Section XIII - The Zeta Function extended to the [ critical strip ]

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

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

original dated 2009-5-08