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 XIV - Riemann's Prime Counting Function F(x) and its derived f(x)

Summary for the Section :

• In this Section , we shall take a quick look at the function F(x) , Riemann's { Prime Counting Function } ,
  • to arrive at its derived function f(x) .

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Riemann's Prime Counting Function :

• In his paper of 1859 , Riemann defined the Prime Counting Function [ F(x) ] as follows :
  • For the value of [ x ] not being equal to a { prime number } :
    • 

  • For the value of [ x ] being equal to a { prime number } :
    • 

      

  Very simply , F(x) is a { step-function } with a jump whenever the value of [ x ] is a { prime number } .

• And we take note , at this point , of these properties for F(x) :
  • { F(x) = 0 } for { [ x ] being less than [ 2 ] } ,
    • arising from the fact that [ 2 ] is the first-and-smallest { prime number } ;

  • each jump at a { prime number } adds [ 1 ] to the value of Function F(x) .

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The Value of { log [ Zeta-of-(s) ] } :

• We note here that Riemann then used this definition for the { Zeta Function } for the developments to follow :
  • 

    

  We can then re-write this equation in this format for easier munipulations :

  • 

• Let us now [ take-the-log ] on both sides of the equation , and we have :
  • 

    yielding :

  • 

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Taylor Series Expansion :

• Let us now bring-in the Taylor Series expansion formula for [ Log (-x) ] , i.e :
  • 

  Applying this formula to the equation immediately above then yields us this :

  • 

• We can then re-write the above equation in this format , for the munipulations to follow :
  • 

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The Reciprocals :

• We now bring-in , as Riemann did in 1859 , these equations for the { reciprocals } , i.e. :
  • 

  • 

  • 

    And so-on-and-so-forth .

  Substituting these into the equation we have immediately above then yields us the following :

  • 

    

  And on dividing both sides by [ s ] , we have :

  • 

    

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The Interesting SWAP :

• Here comes the interesting part , namely :
  • Riemann then swapped the order of the { Summation Signs } and the { Integral Signs } ,
    • to come up with this equation :

  • 

    where the function f(x) is defined by :

    • 

  And everything does seem proper at first glance .

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An Interesting Property of the function f(x) :

• We take note here again that :
  • the function F(x) is a { step-function } with the [ steps / jumps ] occuring :
    • whenever the value of [ x ] hits a { prime number } .

  As such ,

  • the derived function f(x) is also a { step-function } with the [ steps / jumps ] occuring :
    • whenever the value of [ x ] hits a { prime number } or a { power-of-prime } .

• Let us bring back , at this point , the L.C.M.-based function [ Lamda-of-(M) ] from Section III :
  • 

    or alternately , this format for evaluation purposes from Section IV :

  • 

    where the [ Q-i ]'s are governed by :

    • 

  And we take note here again that the [ Lamda-of-(M) ] function is a { step-function } with the [ steps / jumps ] occuring :

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

• Then , for the 2 functions [ f(x) ] and [ Lamda-of-(M) ] :
  • is this where the similarity begins or is it where it ends ?

    Another good question , but again remains unanswered !

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go to the next section : Section XV - Finite Values at [ s = 1 ] for a finite value of [ M ]

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

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

original dated 2009-5-08