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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Epilog II - The Separability of Riemann's f(x) Function into its component parts

Summary for this Epilog :

• The development of Riemann's [ f(x) Function ] is based primarily on a Taylor Series expansion procedure ,
  • conducted on a { [ prime-number ]-by-[ prime number ] } basis .

  As such , the [ f(x) Function ] , being the summation result thereof ,

  • is in-itself-and-by-itself separable into its component parts .

• This epilog then gives the finer details on this phenomenon .

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Recalling the development process :

• Let us recall that in his paper of 1859 on { prime numbers } , Riemann started-off with the Euler Observation , i.e. :
  • 

  We can then re-write each individual term on the left-hand-side in this format :

  • 

  And on [ taking-logarithms ] , we have :

  • 

• Let us bring-in , at this point , the Taylor Series expansion formula for { negative log's } , i.e. :
  • 

  On applying this to the equation above then yields :

  • 

• Let us now express the [ reciprocals ] , as Riemann did in 1859 , in this format :
  • 

  • 

  • 

  • and so-on-and-so-forth .

  And on applying these to the equation we have above then yield us this , for an individual term on the left-hand-side of the Euler Observation :

  • 

    

  And on re-arranging , we have :

  • 

    

• Thus , on consolidating the integrands , we can now write the above equation in this format , for each individual { prime number } [ p-i ] :
  • 

    where the [ PHI-i-of-(x) ] function is a { step function } associated with that particular { prime number } [ p-i ] .

  Let us now look at this [ PHI-i-of-(x) ] function more closely , next .

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[ PHI-i-of-(x) ] for [ i = 1 ] :

• We note here that the 1st { prime number } is [ 2 ] .

  Thus , we can write :

  • 

  And the properties for the { step function } [ PHI-1-of-(x) ] are as follows :

  • 

  • 

  • 

  • 

  • 

  • and so-on-and-so-forth .

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[ PHI-i-of-(x) ] for [ i = 2 ] :

• We note here that the 2nd { prime number } is [ 3 ] .

  Thus , we can write :

  • 

  And the properties for the { step function } [ PHI-2-of-(x) ] are as follows :

  • 

  • 

  • 

  • 

  • 

  • and so-on-and-so-forth .

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[ PHI-i-of-(x) ] for [ i = 600 ] :

• We note here that the 600th { prime number } is [ 4409 ] .

  Thus , we can write :

  • 

  And the properties for the { step function } [ PHI-600-of-(x) ] are as follows :

  • 

  • 

  • 

  • 

  • 

  • and so-on-and-so-forth .

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The Separability of the [ f(x) Function ] :

• Let us bring back , at this point , the equation from above for an indvidual term , i.e. :
  • 

• Let us suppose , for the time-being , that we are investigating the { prime numbers } up to [ M ] only .

  And based on our understanding of the [ PHI-i-of-(x) ] { step functions } above , we can now write :

  • 

  yielding :

  • 

    where :

    • 

• Let us now [ take-the-limit ] and let [ M ] go to [ infinity ] , and we have :
  • 

  yielding :

  • 

    where :

    • 

  This then leads us to this equation :

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    and finally yielding :

  • 

• Essentially , we were able to break-down the [ f(x) Function ] into its component parts .

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Setting up the [ UTT-i ] 's for a given value of [ x ] :

• For the purpose of a further evaluation of the [ PHI-i-of-(x) ] functions ,
  • let us now set up the [ UTT-i ]'s for a given and finite value of [ x ] .

  We then define the [ UTT-i ]'s via :

  • 

    [ UTT-i ] is then the quotient of { log [ x ] } over { log [ p-i ] } ;

    • and is named as such , i.e. [ qUoTienT ] .

• Let us now split the [ UTT-i ]'s into its { integer part } and its { fractional part } , as follows :
  • 

    

  And by following closely / almost-exactly the procedure we have outlined for the [ QTT-i ]'s in Section IV ,

  • we can then arrive at this very-similar / almost-exactly-the-same relation for the [ UTT-i ]'s :
    • 

    And this equation tells us that :

    • when we sum up the { reciprocals } for the [ PHI-i-of-(x) ] function up-to-[ x ] ,
      • we need to stop at the value [ U-i ] ;

        otherwise we would have gone beyond [ x ] and brought-in extra-unwanted-terms .

• Thus , we can now write this set of two (2) defining equations for the [ PHI-i-of-(x) ] function :
  • 

    and :

  • 

    noting here that the [ PHI-i-of-(x) ] is , by its nature , the sum of the [ reciprocals ] up to the value [ U-i ] .

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A Brief Comment here :

• Since the [ PHI-i-of-(x) ] functions are { step functions } and only [ jumps / changes values ] ,
  • whenever the value of [ x ] hits the { prime number } [ p-i ] or a { power-of-the-prime } ;

  it then follows that :

  • the [ f(x) Function ] , being a summation of the [ PHI-i-of-(x) ] functions ,

    also only [ jumps / changes value ] :

    • whenever the value of [ x ] hits a { prime } or a { power-of-prime } .

• This then accounts for its striking similarity between the { f(x) Function } here and the L.C.M.-based { Lamda-of-(M) Function } from Section III .

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go to the next epilog : Epilog III - The gap in-between { consecutive prime numbers } to reach Infinity

go to the last epilog : Epilog I - Using { L.C.M.-based series modulo's } in the Fast-Track Approach

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

original dated 2009-5-17 updated 2009-6-02