However we could equally use combinations of the same prime.
Let us take to illustrate the simplest case where 2 is multiplied by 2 to obtain 4!
Now if we consider for example the sum over natural numbers, where only every 4th term is included, this in general terms can be represented as,
1/4s+ 1/8s + 1/12s +.....
And the sum of this series is simply given by the standard product expression (for all primes) multiplied by 1/4s.
So for example when s = 2,
1/42+ 1/82 + 1/122 +..... = 1/42(4/3 * 9/8 * 25/24 *...) = π2/96.
And then of course each term (in the sums over natural numbers) need not occur just once!
For example let us now decide to subtract four times the previous series (just dealt with) from the original zeta series, so that we obtain,
1/12 + 1/22 + 1/32 – 3/42 + 1/52 + 1/62 + 1/72 – 3/82 +...
Now the sum of this series = π2/6 – 4(π2/96) = π2/6 – π2/24 = π2/8.
And as this series has the same sum as that over the odd natural numbers, the corresponding product over primes expression = 9/8 * 25/24 * 49/47 * ....
Now just as with the well-known Zeta 1 (Riemann) function, it should then be possible to extend all the reduced functions in similar fashion (through analytic continuation) to all values in the complex plane (except 1).
This would thereby entail a unique functional equation for the reduced functions (whereby the value of each function for values of s > 1, can be mapped with corresponding values for 1 – s).
It would also entail that a corresponding Riemann type hypothesis would exist for each of the reduced functions, with all the zeros for each function lying on the imaginary line through .5.
Now one might speculate as to what the precise function of all these zeros for each of the reduced functions might be!
Basically, it provides in each case the means to exactly predict the number of primes (for a given set of numbers) dictated by the precise nature of the reduced function.
For example, if we omit all numbers divisible by - say - 2 and 3 - we then obtain the following reduced zeta function,
ζ(s) = 1/1s + 1/5s + 1/7s + ....
Then for values of this (reduced) function, where ζ(s) = 0, s will take on a unique set of complex values of the form .5 + it and .5 – it respectively.
Because all numbers containing 2 and 3 as factors are omitted, this entails that the set of natural numbers arising will be considerably smaller (as only the remaining primes can be factors).
Then with respect to prime frequency, we are now faced with the task of predicting the exact number that might occur up to n (where n is defined with respect to the combined frequency of those numbers (where 2 and 3 are not factors).
Thus as we reduce the number of permissible primes, the corresponding number of composites arising will drop rapidly.
So as we have seen earlier, where 2 and 3 are omitted as factors, we only have 3 instances up to 50 of composite numbers involving combinations of other primes i.e. 25, 35 and 49.
Now, one might well initially question the value of these reduced functions, as the frequency of primes (to a given number) can perhaps more easily be obtained with reference to the zeros of the standard (Zeta 1) function containing all primes as factors.
However, as repeatedly stated on previous occasions, properly interpreted, the Zeta 1 function always operates in complementary fashion with the corresponding Zeta 2 function (both of which are understood in a dynamic interactive manner).
And just as the Zeta 1 gives rise to a potentially unlimited number of reduced functions, in complementary fashion, the Zeta 2 likewise gives rise to an unlimited set of corresponding enhanced
functions. And it is only in the two-way interaction of both functions that we can properly understand the relationship of the primes to the natural numbers (both externally and internally).
So it is to these enhanced Zeta 2 functions that we will next turn!.