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

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

Top Of Page

Section IV - Setting up the [ QTT-i ]'s for any given value of [ M ]

Summary for the Section :

• In this Section , we shall set up , for any given and finite value of [ M ] :
  • the values of the [ QTT-i ]'s , with each [ QTT-i ] being correspondent to a particular and specific { prime number } [ p-i ] .

go to Top Of Page

Defining the values of the [ QTT-i ]'s :

• For any given and finite value of [ M ] , greater than or equal to [ 2 ] :
  • we now define , for each and every { prime number } [ p-i ] , a corresponding value [ QTT-i ] ,
    • such that :
      • 

    [ QTT-i ] is then the quotient of [ log M ] divided by [ log p-i ] ;

    • and hence the name [ QTT-i ] , abbreviated from the word [ QuoTienT ] .

go to Top Of Page

Splitting [ QTT-i ] into its { integer part } and its { fractional part } :

• Let us now split each of the [ QTT-i ]'s into its { integer part } and its { fractional part } , via :
  • 

    

  And the { fractional part } is , as always , governed by its defining relation , i.e. :

  • 

• Let us now recall the original definition for the [ QTT-i ]'s from above , i.e. :
  • 

  Thus, we can now write this new relation :

  • 

go to Top Of Page

Deriving the Bread-And-Butter Inequality :

• We start-off this derivation with , again , the original definition for the [ QTT-i ]'s :
  • 

  We can then write :

  • 

    yielding :

  • 

  And consequently , we also have :

  • 

• Let us now bring-back the defining relation for the { fractional part } of [ QTT-i ] , i.e. :
  • 

  As such , we can now write this inequality :

  • 

    And this equation is valid as long as :

    • the { prime number } [ p-i ] remains a { positive integer } , and
    • the value of [ Q-i ] is { greater than or equal to [ zero ] } .

• Thus , substituting the equation we have above for [ M ] , i.e. :
  • 

    into the inequality immediately above , we can now write :

  • 

  And this is our Bread-And-Butter Inequality , ready for our use in the next Section and beyond .

go to Top Of Page

A Special Note here :

• And in preparation for the next Section , we shall now take special note here of the following :
  • Whenever [ M ] is a { prime number } or a { power-of-prime } , this relation below is true and valid :
    • 

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

    And the above Bread-And-Butter Inequality , under such circumstances , can be split into two (2) separate inequalities , namely :

    • 

      and :

    • 

  • Otherwise , whenever [ M ] is neither a { prime number } nor a { power-of-prime } , this relation below is always true and valid :
    • 

    And the above Bread-And-Butter Inequality , under such circumstances , becomes :

    • 

      for all the [ p-i ]'s .

go to Top Of Page

Calculation for the value of [ Lamda-of-(M) ] :

• Let us bring back , at this point , the definition for the function [ Lamda-of-(M) ] :
  • 

  The calculation of the [ Lamda-of-(M) ] must , therefore accordingly , follow along these guide-lines :

  • { natural numbers from [ 2 ] to [ M ] } are not divisible by { [ prime numbers ] greater than [ M ] } ,
    • and these [ prime numbers ] are not involved in the L.C.M. calculations here .

  • [ prime factors ] of { natural numbers from [ 2 ] to [ M ] } can only involve { [ prime numbers ] less than or equal to [ M ] } ;
    • and the L.C.M. must contain each-and-everyone of these said-[ prime numbers ] ,
      • in order that all { natural numbers from [ 2 ] to [ M ] } are fully covered in the L.C.M. .

  • the L.C.M. is therefore the product of these said-[ prime numbers ] raised-to an appropriate-power ,
    • and the appropriate-power is governed by the our bread-and-butter relation above , i.e. :
      • 

        where [ Q-i ] is the { appropriate-power } for the { prime number } [ p-i ] .

    And that is to say :

    • It is only necessary to include { [ p-i ] raised-to-the-power [ Q-i ] } ,

    • but it is inappropriate / unnecessary to include { [ p-i ] raised-to-the-power [ Q-i + 1 ] .

• As such , we can now write :
  • 

  And this tell us that the value of the L.C.M. only changes ,

  • whenever the value of [ M ] hits a [ prime number ] or a [ power-of-prime ] ,
    • thereby causing a change in the associated [ Q-i ] for that particular [ prime number ] or [ power-of-prime ] .

go to Top Of Page

go to the next section : Section V - The Value { [ M ] raised to the power [ pi-of-(M) ] }

go to the last section : Section III - Setting up the L.C.M.-based Function [ Lamda-of-(M) ]

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

original dated 2009-5-08