Relating Sum over Natural Numbers with Products over Primes

A necessary feature of all L-functions is that they can be expressed as infinite expressions, where a sum over the natural numbers is equal to a corresponding product over the primes for real exponents of s (s > 1).

For example with respect to the Riemann zeta function (which is the best known) an infinite sum over all the natural numbers is equal to an infinite product over all the primes.

     _∞

So ∑1/n^s    =      ∏1/(1 – 1/p^s)

    ^{n = 1                        p}

So when for example s = 2,

1/1² + 1/2² + 1/3² + 1/4² + …       = 4/3 * 8/7 * 25/24 * 49/48 * …   = π²/6

However the key significance of this relationship is missed from the conventional mathematical perspective, where the equations are interpreted in a merely analytic (quantitative) manner.

In fact, properly understood both expressions are dynamically interrelated in a relative manner with twin analytic (quantitative) and holistic (qualitative) aspects, which operate in a complementary fashion.

If for a moment, we concentrate on the first term of the product over primes expression i.e. 4/3, we can view this individual term in an analytic (quantitative) manner, which is directly related to the corresponding first prime i.e. 2.

However this prime relates to the corresponding sum over natural numbers expression in a complementary holistic (qualitative) manner.

So rather than being independent, 2 as a prime is related to all even composite natural numbers as a shared factor. So we move from the individual quantitative notion of 2 (as an independent number) to the collective qualitative notion of that same prime as a shared factor (and thereby interdependent with other factors) with respect to all even natural numbers.

So again when we view each individual prime (in the product over primes expression) in an analytic (quantitative) manner, then in relative terms we must view each corresponding prime (with respect to the sum over natural numbers expression) collectively in a holistic (qualitative) fashion. Here it is seen in each case as a shared factor of a potentially unlimited set of composite natural numbers.

However when we now view the product of all primes with respect to the right hand expression, the frame of reference shifts so that the collective multiplication of these primes in complementary manner now attains a holistic (qualitative) meaning.

Thus the same dynamics are at work here that led us to earlier see that the multiplication of 1 by 1 i.e. 1¹ * 1¹  = 1²_, entails, relative to addition, the holistic qualitative notion of 2. Thus the collective multiplication of the prime related terms changes interpretation from quantitative and thereby independent (in each individual case) to qualitative and thereby relatively interdependent (in the collective situation).

And again in complementary fashion, when we view the addition, in the left hand expression of all the independent natural number terms, this entails the analytic (quantitative) notion of number.

And as always in dynamic interactive terms, the terms of reference can switch so that we can equally view a prime factor such as 2 in independent terms, whereby it attains a direct quantitative significance. However, in this case, the corresponding term in the product over primes expression i.e. 4/3, now acquires a holistic meaning in a relative complementary manner (where the unit members of 2 are understood as interdependent). 

And finally the collective product of all prime terms in the right hand expression can be likewise given a quantitative meaning, with again in relative complementary fashion, the collective sum of the natural number terms (in the left hand expression) now acquiring by contrast a qualitative interpretation.

So the key point is that all number terms both individually and collectively in both expressions can be given dual analytic (quantitative) and holistic (qualitative) interpretations.

And the relationship between the two expressions is of a dynamic interactive nature in a complementary relative manner, where analytic is balanced by holistic and holistic balanced by analytic interpretation respectively.

However because in isolation all number terms can indeed be given an independent quantitative meaning, this leads in conventional mathematical interpretation to the reduction of holistic meaning (with respect to the various number terms) in an absolute quantitative manner.  

As we have seen in our example above ζ(2) = π²/6.

And on careful reflection this very value clearly reflects the dynamic relative nature of number with respect to the Riemann zeta function, where both analytic (linear) and holistic (circular) aspects of understanding interact in two-way fashion with each other.

Again in conventional mathematical terms, the crucially important infinite notion itself becomes reduced in a merely finite linear manner.

So for example with respect to the sum over natural numbers expression, when we add a finite number of terms the sum is a rational number.

Thus again with respect to ζ(2), 1/1² = 1;1/1² + 1/2²  = 5/4; 1/1² + 1/2²  + 1/3²  = 49/36 and so on.

Likewise when from the standard perspective we multiply a finite number of rational terms, we obtain another rational number.

Thus no matter how many finite rational terms are added (or multiplied) in an analytic (quantitative) manner, the sum (or product) is a rational number.

However the fact that the infinite sum is transcendental (and not rational) clearly suggest that a qualitatively distinct notion is required in moving from finite to infinite with respect to the zeta function.

This strongly suggests in turn that very relationship as between finite and infinite is of a relative - rather than absolute - nature.

Thus the reduced notion that somehow the infinite notion can be approached through the successive adding (or multiplying) of finite terms in a linear quantitative manner is without foundation.

So when one starts by defining finite notions in an analytic (linear) manner, then the infinite notion - by contrast - relates to corresponding holistic (circular) interpretation.

Now once again holistic appreciation is directly of an ineffable formless nature; however indirectly in can then be expressed in paradoxical (i.e. circular) rational manner.

When one reflect on the nature of π - which is the most important of the transcendental numbers - from the standard quantitative perspective, it reflects the relationship of its (full) circular circumference to its line diameter.   

Then when one reflects equally on the nature of π from the (unrecognised) holistic qualitative perspective it reflects the dynamic relationship of both pure circular (qualitative) and linear (quantitative) understanding.

Now in my interpretation of  ζ(2), I have attempted to show the nature of such transcendental interpretation from a holistic (qualitative) perspective.

So the holistic nature of transcendental understanding - directly mirrored in this case by the nature of π - is to perfectly balance linear (quantitative) with circular (qualitative) understanding.

Then in this manner the quantitative result of each expression is matched by an appropriate qualitative interpretation of the relationships involved.

It is well known that for all even integer values of ζ(s) where s > 1, that the quantitative results for both the sum over natural numbers and product over primes expressions = kπ^s, where k is a rational number.   

This reflects the fact that the even roots of 1 can always be arranged in a complementary manner, where for every complex root a corresponding negative root will exist.

The fact that such complementarity does not strictly hold for odd numbered roots provides the holistic explanation as to why π is not involved in quantitative results for odd integer values of
ζ(s).