Zeta Zeros Made Simple (11)

We have seen that in holistic terms, the various roots of 1 provide the appropriate means of expressing the unique ordinal nature of each number in a group (indirectly in a quantitative manner).

So once again, for example, in a prime group of 5 members, the 5 roots of 1, i.e. 1^1/5, 1^2/5, 1^3/5, 1^4/5, 1^5/5 holistically express the notions of 1st, 2nd, 3rd, 4th and 5th in the context of 5 members.

And of course the final root 1^5/5 = 1, expresses the default case of the 5th (in the context of 5), which directly relates to the last unit of 5, just as 1^4/4 represents the last unit of  4, 1^3/3 the last unit of 3, 1^2/2 the last unit of 2 and 1^1/1 the last unit of 1 respectively.

And these holistic expressions relate directly to a circular - rather than linear - notion of number, that geometrically is represented as the equidistant points on the unit circle (in the complex plane).


In conventional Type 1 terms, the primes are unique in that they provide the basic building blocks from which all other composite natural numbers are derived (in a quantitative manner) .

However, here in reverse complementary manner, we can see the alternative nature of primes, in that each prime is composed of a unique set of natural number roots (which cannot be replicated for any other prime).

Now strictly this statement needs qualification, in that one root i.e. 1^n/n is always non-unique as 1.
In fact, in this context, we could refer to this default root as the trivial root of 1.

So the t roots of 1 are expressed as 1 = s^t, i.e. 1 - s^t  = 0.

Therefore to find the expression for the non-trivial roots, we divide the expreby the trivial root i.e. 1 - s = 0.

This then yields the expression 1 + s¹ + s² + s ³ + ..... + s^{t – 1} = 0.

I refer to this as the Zeta 2 Function, which complements the Riemann (Zeta 1) Function,

i.e. 1 ^{– s}  + 2 ^{– s} + 3 ^{– s} + 4 ^{– s} + ......   = 0

Whereas the Zeta 1 Function is infinite, the Zeta 2 function is finite; then whereas the natural numbers appear as base numbers with respect to the Zeta 1, they appear as dimensional numbers with respect to the Zeta 2;  finally whereas the unknown s appears as a (negative) dimensional number in the Zeta 1, it appears as a (positive) base number in the Zeta 2!


If we initially, with respect to the Zeta 2, confine the value of t to the primes, then the solutions for s represent all the unique ordinal natural number solutions for s.

For example when t = 3, the Zeta 2 Function is

1 + s¹ + s²   = 0.

Therefore the 2 unique solutions (i.e. non-trivial zeros) in this case are

– .5 + .866i and – .5 –  .866i respectively.

And these solutions represent the indirect quantitative expression of the qualitative notions of 1st and 2nd (in the context of 3 members). Because quantitative and qualitative notions are thereby related, these properly require holistic mathematical understanding.

And these solutions  which uniquely express ordinal notions (in the context of different sized groups), I refer to as the Zeta 2 zeros.

So each non-trivial natural numbered root (within a prime group), can be given a unique quantitative expression (in a relative manner).

However - by definition - the sum of all roots (including trivial) = 0.

What this entails, in holistic mathematical terms, is that the collective sum of roots has no quantitative significance.

So notice the complementarity!

In cardinal Type 1 terms 1 + 1 + 1 = 3. Here, though the collective sum of components has a quantitative significance, each of the individual component parts (as homogeneous) has no qualitative significance.

Then in Type 2 terms 1st + 2nd + 3rd,  indirectly represented quantitatively as  – .5 + .866i  – .5 –  .866i + 1= 0.

Here though the individual components have an (indirect) quantitative significance, the collective sum has no quantitative significance. In other words it has a merely qualitative  interdependent significance (where all arbitrary relative connections as between quantitative variables cancel out).