Zeta 1 and Zeta 2 Functions - Complementary Nature

 Once again I define the Zeta 1 (Riemann) Function as

 ζ₁(s₁) = 1^{– s1}  + 2^{– s1}  + 3^{– s1}  + 4^{– s1}  +…

I then define the Zeta 2 in complementary like fashion as,

ζ₂(s₂) = 1 + s₂¹ + s₂² + s₂³ + s₂⁴ +….

Now I wish to demonstrate an important feature of these two related functions by putting in the first few values for s₁ =0, 1, 2, 3, 4, 5,.....

 ζ₁(0) =  1  +    1     +   1       +   1        +….                            = ∞
 ζ₁(1) =  1  +  1/2    + 1/3     +  1/4      + …                            = ∞
 ζ₁(2) =  1  +  1/4   +  1/9    +  1/16     + ....      = π²/6    =  1.64493…
 ζ₁(3) =  1 +  1/8  + 1/27   +  1/64     + ....                     =  1.20205...
 ζ₁(4) = 1 + 1/16 + 1/81   + 1/256    + ...         = π⁴/90  =  1.08232…
 ζ₁(5) =  1 + 1/32 + 1/243 + 1/1024  + ...                      =  1.03692…
  …………………………………………………………….

Now if instead of reading across each row horizontally, one now reads down each column vertically, the numbers  all conform to equivalent values of the Zeta 2 function.

The 1st column      = ζ₂(s₂), where s₂ = 1,
The 2nd column    =  ζ₂(s₂), where s₂  = 1/2
The 3rd column    =  ζ₂(s₂),  where s₂  = 1/3
The 4^th column    =   ζ₂(s₂), where s₂  = 1/4, and so

In other words when s₂ = 1, where s₁ = 0, and s₂ = 1/s₁ for all other positive integer values of s₁, the horizontal rows representing the expansion of  ζ₁(s₁) for the various values of s_1, exactly match the corresponding columns representing the expansion of ζ₂(s₂) for the various values of s_2.

Put another way, the two functions thereby provide exactly the same information (as viewed from two complementary perspectives).  


In deeper philosophical terms, this entails a continual dynamic interaction with respect to the number system as between two aspects that are whole and part with respect to each other.

So from one perspective, we can attempt to show how the primes and are related to the “whole” number system, with each natural number (other than 1) representing a unique product of prime factors.

And it is in this context that the Zeta 1 (Riemann) function is widely used.

However from the equally valid opposite perspective, we can likewise attempt to show within each "part" number, a Zeta 2 relationship as between prime and natural numbers which now in “Alice in Wonderland” like fashion, is diametrically opposite to the first case.

 However this requires truly again looking at the relationship between primes and natural numbers interactively in a true bi-directional fashion where both analytic (quantitative) notions of independence and holistic (qualitative) notions of interdependence are equally recognised.

So from the initial quantitative perspective, the primes appear collectively as the “building blocks” of the natural number system (in a cardinal manner).

However equally from the - greatly neglected - qualitative perspective, each individual prime now appears as defined by a unique set of natural numbers (in an ordinal manner). So for example 3 is prime which is uniquely defined by its 1^st, 2^nd and 3^rd members in a (qualitative) ordinal manner.

And again from a proper dynamic perspective quantitative notions of number independence and qualitative notions of number interdependence have no strict meaning in isolation from each other.

In like fashion, proper understanding of the key two-way relationships as between prime and natural numbers have no strict meaning without the mutual incorporation of both Zeta 1 and Zeta 2 functions.


However, before going in the next blog to demonstrate the great practical use of the Zeta 2 function in understanding the behaviour of the individual primes, I will attempt to demonstrate here a simple way of proving an important relationship that I long suspected to hold.

So if one subtract 1 in turn from the results of the Zeta 1 (Riemann) function where s = 2, 3, 4, 5, … n (where n is unlimited in size) and then obtains the sum of this series, the result (in the limit) = 1.

So illustrating with the first 8 values,

ζ₁(2) – 1 = .64493…
ζ₁(3) – 1 = .20205…
ζ₁(4) – 1 = .08232..
ζ₁(5) – 1 = .03692…
ζ₁(6) – 1 = .01734…
ζ₁(7) – 1 = .00834…
ζ₁(8) – 1 = .00407…
ζ₁(9) – 1 = .002008..

So ∑ ζ₁(s₁) from s = 2 to s = 9, = .997978, which is already very close to 1.

However bearing in mind, what has been said above, ζ₁(s₁) and ζ₂(s₂) provide identical information where s₁ = 0 and s₂ = 1, respectively and for all other positive integers of s₁, where s₂ = 1/s₁.

Now for s₁ = 0 and s₂ = 1, both zeta functions generate the same infinite series i.e. 1 + 1 + 1 + 1 +…..
So we remove this from both functions.

And as likewise ζ₁(s₁)  is infinite for s₁ = 1, we remove this from consideration. This likewise means that we remove the first value for s₂  in the Zeta 2 function.

And likewise to make ζ₁(s₁) – 1 equivalent with ζ₁(s₂) we need to remove the first value of 1 in each case in the Zeta 2.

Therefore we now have ζ₁(s₁) – 1 defined for s₁ ≥ 2 which is equivalent to s₂² + s₂³ + s₂⁴ +…. defined for s₂ = 1/s₁.

So for this latter (modified) Zeta 2 function the first sum of series for s₂ = 1/2,  = 1/4 + 1/8 + 1/16 + …  = 1/2.

The next sum, for s₃ = 1/3, = 1/9 + 1/27 + 1/81… = 1/6

Continuing on in this manner, for further values of s₂ = 1/4, 1/5, 1/6, and so on, we generate results 1/12, 1/20, 1/30,… and so on.

So the combined sum of values for this Zeta 2 series,
= 1/2 + 1/6 + 1/12 + 1/20 + 1/30 + ….

Now when one looks closely here, certain fascinating number patterns become evident.

The denominator of each term of the combined series can be represented as n * (n + 1).
So 2 = 1 * 2, 6 = 2 * 3, 12 = 3 * 4, 20 = 4 * 5, 30 = 5 * 6, and so on.

And then the sum of terms is then successively given as n/(n + 1).

So 1/(1 * 2) = 1/2; then 1/2 + 1/(2 * 3) = 2/3; 2/3 + 1/(3 * 4) = 3/4; 3/4 + 1/(4 * 5) = 4/5, and so on.

Therefore in general terms the combined sum for n terms of the series = n/(n + 1) and as n increases without limit n(n + 1) = 1.

Thus in like manner, since both are equivalent,  ∑{ζ₁(s₁) – 1) = 1, for s₁  = 2 to ∞.