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 II - A Special Note on the Notations in this paper

Summary for the Section :

• In this Section , we shall list out the primary-and-common { sets / limiting-values / functions / variables } to be used throughout the paper , namely :
  • the set [ Set P ] , being the set of all { prime numbers } ,
  • the value [ M ] as the limiting-value for our investigation for the time-being ,
  • [ pi-of-(x) ] as the { prime number counting function } , and
  • the general format for expressing a { complex number } .

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The Set of { Prime Numbers } :

• Throughout the paper, we shall denote the set of all { prime numbers } as the set [ Set P ] , i.e. :
  • 

• And we shall assume , for our analysis purposes here , that we were able to arrange the { prime numbers } on an ascending-order basis , so that :
  • each and every { prime number } may be assigned an unique-and-precise index [ i ] for identification .

  And that is to say , for any { prime number } denoted by [ p-i ] , i.e. :

  • 

    we have :

    • 
    • 
    • 
    • and so-on-and-so-forth .

  We then list-out the first 100 { prime numbers } for the reader's quick reference :

  Value of [ i ]	Prime Number ( p-i )
  **************	*****	*****	*****	*****	*****	*****	*****	*****	*****	*****
  i = 1 thru 10	2	3	5	7	11	13	17	19	23	29
  i = 11 thru 20	31	37	41	43	47	53	59	61	67	71
  i = 21 thru 30	73	79	83	89	97	101	103	107	109	113
  i = 31 thru 40	127	131	137	139	149	151	157	163	167	173
  i = 41 thru 50	179	181	191	193	197	199	211	223	227	229
  i = 51 thru 60	233	239	241	251	257	263	269	271	277	281
  i = 61 thru 70	283	293	307	311	313	317	331	337	347	349
  i = 71 thru 80	353	359	367	373	379	383	389	397	401	409
  i = 81 thru 90	419	421	431	433	439	443	449	457	461	463
  i = 91 thru 100	467	479	487	491	499	503	509	521	523	541

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Setting up the limiting-value [ M ] :

• We shall now set-up the value [ M ] , a { positive integer } greater than or equal to [ 2 ] , i.e. :
  • 

    as the limiting-value for our investigation , for the time-being .

  And that is to say , in our investigation of { prime numbers } :

  • We shall first focus on all natural numbers less than or equal to [ M ] ,
    • and establish the relations , if any , between the prime and non-prime numbers up to this limiting-value [ M ] .

  • Once we have established the relationships , if-any-at-all ,
    • we shall then let the value of [ M ] move-out towards [ infinity ] ,
      • and make further observations / draw further conlusions , as appropriate .

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The { prime counting function } [ pi-of-(x) ] :

• We shall now define the function [ pi-of-(x) ] as the { prime number counting function } , i.e. :
  • 

  This is in-line with the general notation formats used in contemporary mathematics literature .

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Format for expressing { complex numbers } :

• For any { complex number } [ s ] , this may be expressed as :
  • 

    

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

go to the last section : Section I - Introduction and { a-priori Assumptions }

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

original dated 2009-5-08