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 XI - The Euler Product Formula via the Fung Inequality

Summary for the Section :

• In this Section , we shall take a closer look at the { Euler Product Formula } , i.e. the formula :
  • 

  We shall also introduce / propose an { Inequality } :

  • 

    or alternately :

  • 
    • 

    labeled as the { Fung Inequality } , for-the-time-being , for identification purposes .

go to Top Of Page

Derivation of the { Fung Inequality } :

• For our proposed Inequality , we shall attempt to derive this :
  • by first looking at an evaluation of the 1st , 2nd , and 4th functions from the last Section , for [ M = 31 ] .

go to Top Of Page

Expansion of the [ EZM-of-(s) ] function for [ M = 31 ] :

• Let us recall the definition of the [ EZM-of-(s) ] function from the last Section :
  • 

  On expansion for [ M = 31 ] , we have :

  • 

    

    

    

• There are then 31 terms in this expansion here .

go to Top Of Page

Expansion of the [ RVM-of-(s) ] function for [ M = 31 ] :

• Let us recall the definition of the [ RVM-of-(s) ] function , also from the last Section :
  • 

  On expansion for [ M = 31 ] , we have :

  • 

  This then yields this equation :

  • 

    

    

• We take note , at this point , that in the above equation :
  • the 1st bracket has 5 terms ,
  • the 2nd bracket has 4 terms ,
  • the 3rd bracket has 3 terms ,
  • the 4th thru 11th brackets each has 2 terms ;

    so that , we would expect to have , on full expansion :

    • [ 5 x 4 x 3 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 ] terms, or 15,360 terms .

  And indeed , on full expansion , we have :

  • 

    

    

    

    

• We note here that :
  • first-of-all :
    • the { 31 terms } of the [ EZM-of-(s) ] function for [ M = 31 ] are fully covered by the { 15,360 terms } here .

  • secondly , for { [ s ] being [ real and positive ] } :
    • the { 31 terms } above and the { 15,360 terms } here are also all [ real and positive ] .

  As such , we can now write :

  • 
    • 

• And for the more general case , we can now write this equation / inequality :
  • 
    • 

  And the derivation here can follow along-the-same-lines as we have just outlined above .

go to Top Of Page

The smallest of the { 15,360 terms } :

• Let us now take a look at the smallest of the { 15,360 terms } ,
  • 

  And this smallest term is given by :

  • 

    or simply :

  • 

    And the smallest term here is always given by the product of the { smallest term in each of the 11 brackets } in the original equation .

  We note here that the number in the { denominator } above is in fact the { L.C.M. of natural numbers from [ 2 ] to [ 31 ] } :

  • 

    And this is always the case because of our specific design of the [ RVM-of-(s) ] function .

go to Top Of Page

Expansion of the [ EZ-Lamda(M)-of-(s) ] function for [ M = 31 ] :

• Let us bring-back the 4th function from the last Section :
  • 

  Let us now do an expansion of this function for [ M = 31 ] and we have :

  • 

    

    

    

  • 

  Two (2) things to note here :

  • on the { total number of terms } :
    • 

  • on the smallest term here , this is again given by :
    • 

• We note here that :
  • first-of-all :
    • the { 15,360 terms } of the [ RVM-of-(s) ] function for [ M = 31 ] are fully covered by the { 72 2017 7644 6800 terms } here .

  • secondly , for { [ s ] being [ real and positive ] } :
    • the { 15,360 terms } above and the { 72 2017 7644 6800 terms } here are also all [ real and positive ] .

  As such , we can now write :

  • 
    • 

• And for the more general case , we can now write this equation / inequality :
  • 
    • 

  Again, the derivation here can follow along-the-same-lines as we have just outlined above .

go to Top Of Page

Consolidating the equations :

• On consolidating this Inequality immediatly above with the previous one , we can now write :
  • 
    • 

    And we shall refer to this as the { Fung Inequality } , for-the-time-being , for identifcation purposes .

• And the expanded format here is then :
  • 
    • 

go to Top Of Page

The Euler Product Formula :

• Let us now take a closer look at the equation / inequality immediately above :
  • As we permit the value of [ M ] to move towards [ infinity ] ,
    • we should be able to arrive at the { Euler Product Formula } , i.e. :
      • 

  And this is because :

  • As long as the left-hand-side and the right-hand-side of the { Fung Inequality } , both being { Zeta type functions } ,
    • converge to the same [ finite value ] ;

    the term in-the-middle , being bounded to-the-left and to-the-right , must also converge to the same limit .

• But this is , nonetheless , an interesting and intriguing situation we have just created ! More on this later .

go to Top Of Page

The Special Case of [ M = 2 ] :

• For the Special Case of [ M = 2 ] , we have :
  • 
    • 

  And on full expansion , this then becomes :

  • 

• Noting here that for [ M = 2 ] :
  • 

    we come up with this rather simple equation :

  • 
    • 

  But this is indeed a most-important equation , explained as follows :

  • For the { Fung Inequality } , the three (3) functions involved are :
    • the [ EZM-of-(s) ] function ,
    • the [ RVM-of-(s) ] function , and
    • the [ EZ-Lamda(M)-of-(s) ] function ;

      and these 3 functions always meet and share the exact-same [ complex value ] at [ M = 2 ] ,

      • 

    The question then arises :

    • As we let the value of [ M ] move-out towards [ infinity ] ,
      • will the values of the three (3) functions meet again ,

        or alternately :

      • will they converge towards the same limit , complex or otherwise ?

go to Top Of Page

• Let us now move-on to the Riemann Zeta Function , next , to find out more !

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

go to the last section : Section X - Setting up 3 more functions

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

original dated 2009-5-08