Zeta 2 Function

Though the analytic properties of the circular number system have been extensively investigated, little or no recognition yet exists as to their true holistic significance.

Much is rightly made of the famed Riemann zeta function in studying the relationship between the primes and natural numbers. However, it seems to me that the true importance of a much simpler complementary function has been greatly overlooked.

So if we refer to the Riemann zeta function as the Zeta 1 function, a dynamic complementary function (i.e. Zeta 2) that in analytic terms relates to the various roots of one likewise exists.

And as in essence this second function is inherently simpler to understand, we can learn a lot regarding the true nature of the famed Riemann zeros, from appreciation of the corresponding set of zeros associated with the Zeta 2 function.


I have stated that the true qualitative meaning of ordinal notions such as 1^st, 2^nd, 3^rd, …, inherently requires a proper holistic (potential) appreciation that is not reduced in a cardinal (actual) manner.

In standard analytic terms, the ordinal notion is always fixed with the last member of its number group. So 1^st is the last (and only) member of a group of 1; then with this fixed, 2^nd is the last of a group of 2; 3^rd is then identified as the last of a group of 3 and so on.

In this way the ordinal members can be represented as successive units on the number line.

So if we look for example at the 3 roots of 1, 1^1/3, 1^2/3 and 1^3/3 (– .5 + .866i, – .5 – .866i and 1) the last root, i.e. 1^3/3,  = 1.

Then earlier, with the 2 roots of 1, 1^1/2 and 1^2/2 ( – 1 and  + 1), the last root, i.e. 1^2/2,  = 1.

And with the 1 root of 1, 1^1/1 (+ 1), the last - and only - root, i.e. 1^1/1, = 1.  

Thus 1^st + 2^nd + 3^rd from this analytic perspective = 1 + 1 + 1. So, the ordinal notion of number is thereby successfully reduced in a cardinal manner.


Again, in the context of a prime group of 3 members, the last root i.e. 1^3/3, corresponds with the analytic notion of 3^rd (of 3) However, the other two roots i.e. 1^1/3 and 1^2/3 correspond with the holistic notion of 1^st (of 3) and the 2^nd (of 3) respectively (where positions are interchangeable).

Now in general terms, 1 = xⁿ, so that 1 – xⁿ = 0. However, as we have seen, for the last root of n (which corresponds to the standard analytic interpretation of ordinal numbers),

1 – x = 0.

Therefore, to isolate the remaining holistic solutions, we obtain (1 – xⁿ)/(1 – x) = 0, i.e.

1 + x¹ + x²  + x³  + … + x^{n – 1} = 0.

This is what I refer to as the Zeta 2 function, which is complementary to the Zeta 1 (Riemann) function i.e.

1^{– s} + 2^{– s} + 3^{– s} + 4^{– s} + …, in the following ways.


Whereas the Zeta 1 is infinite in form, the Zeta 2 here is finite in nature (though without limit). However, the Zeta 2 can also be fruitfully extended in an infinite manner, where

1 + x¹ + x²  + x³  + …     = 1/(1 – x).

In the Zeta 1 (sum over the positive integers expression), the base aspect represent the ordered sequence of the natural numbers, whereas with the Zeta 2, the dimensional aspect (as exponent) now represent the ordered sequence of natural numbers.

Whereas the dimensional number s, is negative in the Zeta 1, the corresponding dimensional number (as one of the natural numbers) is positive in the Zeta 2.

And such complementarity readily implies that both functions - which really are two related aspects of the same function - are connected with each other in a dynamic interactive manner!

With reference to the finite Zeta 2 equation above, when n is prime, the set of solutions - which thereby represent the zeros of the function - is by definition unique.


So these solutions - indirectly given as various quantitative roots of 1 - represent the (potential) holistic nature of each individual prime, where a purely relative “circle of interdependence” uniquely attaches to its various members in an ordinal manner.

Thus, if we take a prime number such as 5, the corresponding Zeta 2 equation is,

1 + x¹ + x² + x³ + x⁴ = 0.

The quantitative solutions for this equation - thereby representing its zeros - are by definition unique (and cannot occur for any other prime). These solutions then indirectly represent the holistic qualitative notions of 1^st, 2^nd, 3^rd and 4^th (of a group of 5 members). Then together with the default 5^th member, where the quantitative representation = 1, the sum of these 5 roots = 0.

This, of course, has been long understood from a quantitative perspective.

However the true holistic significance of this finding is that it indirectly represents, in a quantitative manner, how the unique qualitative identity of each specific member is seamlessly integrated, in a relative manner, with the collective qualitative interdependence of all the ordinal members of its number group.

And again, where the number is prime, all natural number ordinal members, except the last - which represents the default analytic interpretation - are, by definition, unique.

So for example with respect to the prime number 5, the 5 roots of 1 indirectly express, in a quantitative manner, the unique qualitative nature (except for the default last root) of its 5 ordinal members.

Then the sum of these roots = 0, indirectly expresses the collective qualitative interdependence of all 5 members. As can be seen here, there is always some inevitable overlap as between the twin notions of independence and interdependence respectively, as they mutually imply each other in dynamic interactive terms.


Thus we can conclude this section by stating that when understood in the appropriate dynamic manner, the Zeta 2 zeros - both for prime and then by extension for composite natural numbers - are clearly seen to represent, in an indirect quantitative fashion, the (potential) holistic nature of the ordinal numbers.

And in dynamic complementary terms, this already points to the true nature of the famed Zeta 1 (Riemann) zeros.

In other words, just as from a dynamic perspective, the Zeta 2 zeros represent the holistic counterpart to the standard analytic interpretation of the ordinal numbers, in like manner, the Zeta 1 (non-trivial) zeros, represent the corresponding holistic counterpart to the standard analytic interpretation of the cardinal numbers (as the products through multiplication of unique prime factors).

But of course this truth cannot be recognised while maintaining a mathematical interpretation that formally remains completely confined to the accepted analytic interpretation of number!

So really, my primary intention here is to strongly challenge the existing mathematical consensus confined in formal terms to a mere analytic interpretation, which represents but a special limiting case of an altogether much more comprehensive understanding.