S15. Finite Values at [ s = 1 ]

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

Section XV - Finite Values at [ s = 1 ] for a finite value of [ M ]

In this Section , we shall take a quick look at values of the [ EZM-of-(s) ] function and the [ RVM-of-(s) ] function :
- for the value of [ s ] being [ 1 ] ;
  - i.e. : the { real number } [ 1 ] .

Let us first recall the definition for the [ EZM-of-(s) ] function :
- and also for the [ RVM-of-(s) ] function :
The values of these for [ M = 31 ] at [ s = 1 ] are then :
- and :
We simply note here that they share the same denominator here .

In general , the [ EZM-of-(s) ] function and the [ RVM-of-(s) ] function always share the same denomintor :
- on the half-plane :
Let us find out why this is so .
For the [ EZM-of-(s) ] function , this should be obvious ,
- Since :
  - so that every [ natural number ] from [ 2 ] to [ M ] is divisible into this [ L.C.M. ] ;
  the value of { [ Lamda-of-(M) ] raised-to the power [ s ] } is therefore the natural nominated [ denominator ] ;
  - before any further reductions arising from additional [ common factors ] found in both the [ numerator ] and the [ denominator ] .
For the [ RVM-of-(s) ] function :
- The original defining equation :
  can be rewritten in this format :
  - yielding :
- Let us now take a quick look at the expansion for [ M = 31 ] using this format , and we have :
  We see here that , under this type of expansion :
  - the value [ Lamda-of-(M) ] is also the natural nominated [ denominator ] .
As such , the [ EZM-of-(s) ] and the [ RVM-of-(s) ] functions always share the same denominator , i.e. :

Since the [ denominator ] is the same for both ,
- we might as well concentrate our efforts on the [ nominators ] !

We notice here that , for the [ LVM-of-(s) ] function :
- the value here for a finite value of [ M ] and [ s = 1 ] is also finite .
Thus , all three (3) functions ,
- the [ EZM-of-(s) ] function ,
- the [ RVM-of-(s) ] function , and
- the [ LVM-of-(s) ] function ;
  
  all exhibited finite values at [ s = 1 ] for a finite value of [ M ] .
As we move from this point into the [ critical strip ] , we would also expect to find finite values :
- for the 3 functions for a finite value of [ M ] .
Is there any special reason why we would want to extend the Zeta Function into the [ critical strip ] ,
- using the procedure we have briefly outlined in Section XIII .
Again , a good question but no answers .