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

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

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Section XIII - The Zeta Function extended to the [ critical strip ]

Summary for the Section :

• In this Section , we shall look at the extension of the Zeta Function to the [ critical strip ] , namely :
  • the section of the { complex plane } where :
    • 

  This will then involve two (2) stages :

  • the 1st Stage involving the munipulation of the { PI function } , and

  • the 2nd Stage being the extension process itself .

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The 1st Stage - Munipulation the { PI function } :

• We start-off the development process by first bringing-in the { PI Function } , namely :
  • 

    ( This is the [ Big PI function ] related to the { Factorial Functions } ) .

  Let us now replace the variable [ t ] in the { integrand } on the right-hand-side by the term [ n * x ] , i.e. we simply let :

  • 

    where [ n ] is treated as if it is constant within the { integrand } , so that :

  • 

• Thus , substituting [ n * x ] in lieu of [ t ] in the { integrand } , we have :
  • 

    yielding :

  • 

  Collecting terms then yield this equation :

  • 

• Since [ n ] and [ z ] may be treated as contant within the { integrand } , we can now write :
  • 

    or ,

  • 

  Let us now use a different { dummy variable } [ s ] , instead of [ z ] , in the above equation , via the relation :

  • 

    and the above equation then becomes :

  • 

• Let us now put this equation into this format :
  • 

    and we are now ready for the next stage in the development process .

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The Second Stage - the Extension itself :

• This next stage here can be rather difficult to follow , and :
  • we shall introduce a new variable [ V ] , for use in the { Zeta-V Function } :
    • 

      with [ V ] here being a { positive integer } in-between [ 2 ] and [ infinity ] .

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

  And summing-up both sides from [ n = 1 ] to [ n = V ] would then give us this equation below :

  • 

    yielding consequently :

  • 

  Let us now put the equation into this format :

  • 

• We notice , at this point , that the terms under the { summation sign } on-the-right is a { geometric series } , i.e. :
  • 

    Applying the general formula for the summation of { geometric series } then yields us this equation :

  • 

• Let us now put this equation into this format :
  • 

    And on consolidating terms , we have :

  • 

• Let us now put this result back into the original equation above , and we have :
  • 

    Re-arranging terms then yield us the same equation in this format :

  • 

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The Equation that Riemann wrote :

• Let us bring-in , at this point , the original equation that Riemann wrote in his 1859 paper on Prime Numbers :
  • i.e. the equation :

    

  We see here this equation is different from the very-last we have above , i.e. the euqation :

  • 

• And to go from this equation to the Riemann equation , we need to get involved with [ taking-the-limit ] , as follows :
  • For the left-hand-side , we have :
    • 

  • For the right-hand-side , we have :
    • 

      and :

    • 

  We simply wish to point out here that :

  • this is where the Riemann Zeta Function first deviated from the more traditional Euler Zeta Function .

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go to the next section : Section XIV - Riemann's Prime Counting Function F(x) and its derived function f(x)

go to the last section : Section XII - Riemann's start-off with the Euler Observation

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

original dated 2009-5-08