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 V - The Value { [ M ] raised-to-the-power [ pi-of-(M) ] }

Summary for the Section :

• In this Section , we shall look at the value [ V ] given by :
  • 

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

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The Importance of the value [ V ] :

• Very simply , once we are able to get-a-fix on the value of [ V ] , the value for [ pi-of-(M) ] is immediately determined .
  • Let us start-off with this defining equation for the value [ V ] :
    • 

  • On taking logarithms for both sides , we have :
    • 

      yielding the simple relation :

    • 

  Rather simple indeed .

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Zeroing-in on the value of [ V ] :

• Let us now bring-back this Bread-And-Butter Inequality from the last Section :
  • 

• Let us now set our focus on the { prime numbers } from [ 2 ] to [ M ] , and we take note that :
  • the [ number of { prime numbers } ] here is given by the value [ pi-of-(M) ] .

  Noting that the Bread-And-Butter Inequality is valid for each and every one of these { prime numbers } , we can now write :

  • 

    with the left-hand-side here being the [ product ] of these { prime numbers [ p-i ]'s raised-to-the-power [ Q-i ] } .

  Thus , we have :

  • 

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The Special Case of [ M = 2 ] :

• For the case of [ M = 2 ] , the above equation is trivial and becomes :
  • 

  And this arises from the fact that :

  • 

    so that the above inequality then becomes , for [ M = 2 ] :

    • 

      which is trivial .

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The General Case of [ M ] being greater than or equal to [ 3 ] :

• For the case of [ M ] being greater than or equal to [ 3 ] , we have :
  • 

  And this is based on these points raised in our discussions in the last Section :

  • In general , this equation below is valid for almost all the [ p-i ]'s :
    • 

    with the only exception here being that :

    • whenever the value of [ M ] hits a [ prime number ] or a [ power-of-prime ] ,

      then this equation below is valid :

      • 

        for that particular { prime number } [ p-j ] in-question .

  Since a minimum two (2) { prime numbers } are always involved whenever [ M ] is greater than or equal to [ 3 ] :

  • The First Equation above , i.e. :
    • 

      is valid , under the circumstances , at least [ one-time ] or [ more than one-time ] ;

  • The Second Equation above , i.e. :
    • 

      can either be valid for [ one-time only ] or alternately [ not-applicable-at-all ] ,

      • depending on whether-or-not [ M ] hits a [ prime number ] or a [ power-of-prime ] .

  As such , when we look at all the terms under the [ product sign ] here ,

  • the [ less-than sign ] will necessarily prevail .

    ( It actually takes only one [ less-than sign ] here to override all the other [ equal-to signs ] ,

    • and that is , if there is any [ equal-to sign ] here at all . )

• Let us now bring back the equation from above , i.e. the equation :
  • 

  For the Left-hand-side , the term here is in fact the L.C.M. , i.e. :

  • 

  For the right-hand side , this may be expressed via the following :

  • 

    yielding :

  • 

    ( The term on-the-very-right is then the { product-of-primes up to [ M ] } ) .

• Thus , we can now write the same equation in this final format :
  • 

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go to the next section : Section VI - Actual Statistics for [ M = 31 ]

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

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

original dated 2009-5-08