Only then can the bi-directional nature of both the primes and natural numbers be properly understood from both quantitative and qualitative perspectives.

When dealing with the Zeta 1 (Riemann) function and more recently with its many possible reduced expressions, we found that what is central in all cases is that these functions can always be expressed in twin fashion as a sum over the natural numbers and a product over the primes.

So with respect to the archetypal Riemann function we have the general relationship

∑ 1/n

^{s }= ∏ 1/(1 – 1/p^{s}), for all values except 1 (where s is a complex variable)
Thus, again, when s for example is real = 2,

1/1

^{2 }+ 1/2^{2}+ 1/3^{2}+ … =^{ }4/3 * 9/8 * 25/24 * ....
Now, we showed in the last couple of blogs how each of the terms in the product over primes is related to the corresponding sum over natural numbers expression.

Thus when for example, we remove 4/3 (related to p = 2) in the product over primes, we then must in corresponding fashion remove all ordinal terms divisible by 2 (i.e. 2nd, 4th, 6th,...) in the sum over natural numbers expression.

Then we remove 9/8 (related to p = 3) in the product over primes, we must likewise remove all ordinal terms divisible by 3 (i.e. 3rd, 6th, 9th,...) in the sum over natural numbers expression.

There is clearly a relationship of collective interdependence involved here with respect to both the product over primes and sum over natural numbers expressions. So each term (4/3, 9/8 and so on) in the product over primes only has meaning in conjunction with all other terms. Likewise the corresponding removal of ordinally linked terms, in the sum over natural numbers, only has meaning in relation to all the remaining terms.

So the Zeta 2 involves a (complementary) inversion of the Zeta 1 function.

Now again in the simple expression n

^{s}, I refer to a as the base and b the corresponding dimensional number respectively.

Therefore, for the sum over natural numbers, in the Zeta 1 function, these are defined with respect to the base (which vary over all the natural numbers). So the dimensional number s, with respect to any given value of the function, remains fixed.

However with respect to the Zeta 2 function, it is in reverse, where now the dimensional number varies over the natural numbers with (for any given value) the base number remaining fixed.

Thus the (infinite) Zeta 2 function is defined as,

ζ(s

_{2}) = 1 + s_{2}^{1}+ s_{2}^{2 }+ s_{2}^{3}+ ...Now, it can be easily shown that the value of this function = 1/(1 – s

_{2}).

For s

_{2}ζ(s

_{2}) = s

_{2}

^{1}+ s

_{2}

^{2 }+ s

_{2}

^{3}+ .

Therefore though subtraction, {ζ(s

_{2}) – s

_{2}ζ(s

_{2})} = 1, i.e. ζ(s

_{2}){(1 – s

_{2})} = 1

So, ζ(s

_{2}) = 1/(1 – s

_{2}).

Therefore when s

_{2 }= 1/p

^{s},

ζ(s

_{2}) = 1/(1 – 1/p

^{s}) = 1 + s

_{2}

^{1}+ s

_{2}

^{2 }+ s

_{2}

^{3}+ ... where s

_{2}= 1/1/p

^{s}.

As we have seen, when s = 2, the first term in the product over primes expression = 1/(1 – 1/2

^{2})

= 1(1 – 1/4) = 1/(3/4) = 4/3.

This term can then be expressed through the Zeta 2 function as a sum over the natural numbers (where the natural numbers now refer to the dimensional - rather than base - aspect of number).

Thus 4/3 = 1 + (1/2

^{2})

^{1}+ (1/2

^{2})

^{2 }+ (1/2

^{2})

^{3 }+

^{... . }= 1 + 1/4 + 1/16 + .... = 4/3.

Therefore the important point to grasp at this juncture is that there are in fact two ways of expressing this relationship underlying the Riemann zeta function i.e. where a sum over the natural numbers is shown to be equal to a product over primes.

In the conventional Zeta 1 formulation, a collective relationship is involved, entailing all terms in both the sum over natural numbers and product over primes expressions respectively.

In the - largely unrecognised - Zeta 2 formulation, an individual relationship is involved, entailing each single term (in the product over primes expression), which is then shown to uniquely relate to a corresponding sum over the natural numbers expression, where the natural numbers now relate to the dimensional powers of each term in the expression (with the base related to a specific prime).

Now, from a merely quantitative perspective we can successfully define this crucial connection as between the sum over natural numbers and product over primes through either the Zeta 1 or Zeta 2 formulations (as both forms mutually imply each other).

However the true key to proper understanding is then to understand both the Zeta 1 and Zeta 2 formulations as complementary, in a dynamic interactive manner.

One then moves from an absolute - merely quantitative - type appreciation to a truly relative appreciation entailing number notions of both relative independence and relative interdependence respectively, in both quantitative (analytic) and qualitative (holistic) terms.

There is another compelling way of demonstrating that both the Zeta 1 and Zeta 2 formulations represent. as it were, but two sides of the same coin.

When we write out for example the Zeta 1 (Riemann) function for positive integer values of s from 0 to 5, we obtain the following,

ζ

_{1}(0) = 1

^{ }+ 1

^{ }+ 1 + 1 +…. = ∞

ζ

_{1}(1) = 1

^{ }+ 1/2

^{ }+ 1/3 + 1/4 + … = ∞

ζ

_{1}(2) = 1

^{ }+ 1/4

^{ }+ 1/9 + 1/16 + .... = π

^{2}/6 = 1.64493…

ζ

_{1}(3) = 1 + 1/8 + 1/27 + 1/64 + .... = 1.20205...

ζ

_{1}(4) = 1 + 1/16 + 1/81 + 1/256 + ... = π

^{4}/90 = 1.08232…

ζ

_{1}(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 = ζ

_{2}(s

_{2}), where s

_{2}= 1,

The 2nd column = ζ

_{2}(s

_{2}), where s

_{2 }= 1/2

The 3rd column = ζ

_{2}(s

_{2}), where s

_{2 }= 1/3

The 4

^{th}column = ζ

_{2}(s

_{2}), where s

_{2}= 1/4, and so

In other words when s

_{2 }= 1, where s

_{1 }= 0, and s

_{2 }= 1/s

_{1 }for all other positive integer values of s

_{1}, the horizontal rows representing the expansion of ζ

_{1}(s

_{1}) for the various values of s

_{1, }exactly match the corresponding columns representing the expansion of ζ

_{2}(s

_{2}) for the various values of s

_{2.}

Thus the horizontal rows correspond to the Zeta 1 function, whereas the vertical columns correspond to the Zeta 2 function.

This points to the all-important observation that there are in fact two related ways for understanding the relationship as between the primes and the natural numbers.

The first way - which completely dominates conventional understanding - is to view it in an external fashion, entailing the collective relationship of the primes as (base) "building blocks" to the natural number system.

And the appropriate way for viewing this external aspect is through the Zeta 1 (Riemann) function, with the frequency of primes (to n) approximated in general terms by the simple formula n/log n.

However there is an alternative internal way of viewing the relationship as between primes and natural numbers as relating to the respective total of factors that each individual member of the number system contains.

For example if we take the number 24 to illustrate, its unique prime factor composition

= 2 * 2 * 2 * 3 so that it contains 2 (distinct) prime factors.

However if we now look at the natural numbers that are divisors of 24, we obtain (excluding the trivial case of 1) 2, 3, 4, 6, 8, 12 and 24. So in this case, 24 as a highly composite number, has 7 distinct natural number factors as divisors.

So the internal relationship of the primes and natural numbers relates to the alternative dimensional aspect of number that gives rise to the factors of each individual number (in prime and natural number terms).

Remarkably the simple relationship between these - now expressing the ratio of natural to prime factors - can be given as n

_{1}/log n

_{1}, where n

_{1 }

_{ }= log n.

Now both of these - like the approach to a crossroads from (separate) N and S directions - can be understood analytically in an unambiguous quantitative manner.

However the key to corresponding holistic appreciation is the ability to understand (as in the case of a crossroads) both directions of approach as complementary in a relative manner.

And just as one is enabled thereby to understand left and right turns at the crossroads as paradoxical, with a merely relative meaning depending on the direction of approach, one is now equally enabled to see the relationship as between the primes and natural numbers, likewise as paradoxical (with merely a relative meaning).

So from an external perspective, the relationship between primes and natural numbers seems to depend on the base characteristics of number (as separate "building blocks").

Then from the internal perspective, the relationship as between primes and natural numbers seems now to depend on dimensional characteristics leading to the notion of number as representing factors (in combined aggregate groupings).

So numbers, both in base and dimensional terms, have a relative independent identity that gives rise to their quantitative identity (when reference frames are considered separately); however equally they have a relative interdependent identity that gives rise to their qualitative identity (when reference frames are considered simultaneously).

And growing appreciation of the holistic nature of this two-way relationship as between primes and natural numbers, ultimately gives rise to the clear realisation that the ultimate origin of the number system is purely synchronous in an ineffable manner, where primes and natural numbers are identical with each other in an ineffable manner.

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