Yesterday, I mentioned how all the individual terms of the Zeta 1 (Riemann) function, for both the (infinite) sum over all the natural numbers and (infinite) product over all the primes, can be expressed in terms of the Zeta 2 function for negative - as well as positive - values of s.
This clearly demonstrates how they are complementary with each other in a dynamic relative manner.
In fact as I am continually demonstrating in these blog entries, one cannot meaningfully attempt to understand the Riemann zeta function (i.e. Zeta 1) in the absence of its complementary Zeta 2 expression.
As is well known, all the numerical values of the Riemann zeta function seem counter-intuitive for s < 0. This demonstrates the stark limitation of the conventional reduced mathematical approach, which attempts to interpret number in a merely absolute quantitative type manner, whereas the true nature of number is dynamically relative, with complementary Type 1 and Type 2 aspects that are quantitative and qualitative with respect to each other.
Thus to understand properly why such counter-intuitive results arise and the appropriate means for interpreting these results, we must again understand the Riemann zeta function in a dynamic interactive manner (with complementary Type 1 and Type 2 aspects).
So in the important context of the Riemann function, Zeta 1 and Zeta 2 expressions represent the complementary interaction of both the Type 1 and Type 2 aspects of the number system.
Much light can be thrown on the underlying significance of this truly relative approach to number by focusing clearly on the two points, where the function remains undefined.
In conventional terms - which again solely recognises the Zeta 1 function (in an absolute manner) - only one such point is highlighted. This occurs where s1 = 1.
Then in the standard analytic (i.e. merely quantitative) manner of interpreting this result, it can be easily demonstrated that the series of terms converges to an infinite result.
Now the deeper holistic reason why this is the case is highly instructive.
As explained before in a previous entry, each term in the Riemann function contains both Type 1 (quantitative) and Type 2 (qualitative) aspects.
Once again, we define the Riemann function (Type 1) in general terms as,
ζ(s1) = 1/1s1 + 1/2s1+ 1/3s1 + 1/4s1 +
Thus if we take for example the function for s1 = 2, in conventional (Type 1) mathematical terms, this will be expressed as
ζ(2) = 1/12 + 1/22 + 1/32 + 1/42 + ... = 1 + 1/4 + 1/9 + 1/16 + ...
Now because these values are all ultimately defined in Type 1 terms (with respect to a default dimensional value of 1), they can be rewritten,
11 + (1/4)1 + (1/9)1 + (1/16)1 + ...
However there is a hidden (unrecognised) Type 2 aspect with respect to all these numbers representing - in complementary fashion - the default base value of 1 (raised to this specific dimensional value of 2).
So if we were to rewrite this series purely in Type 2 terms we would get,
ζ(2) = 12 + 12 + 12 + 12 + ...
Then combining both aspects (Type 1 and Type 2) in relative fashion we obtain,
11 * 12 + (1/4)1 * 12 + (1/9)1 * 12 + (1/16)1 * 12 + ...
So we have a linear (quantitative) Type 1 aspect combined with a circular (qualitative) Type 2 aspect with respect to each term. And it is the relationship between both aspects that is then central to interpreting the structural nature of the numerical result (i.e. π2/6).
However, by definition, when s1 = 1, there is no meaningful Type 2 aspect that can be provided, as this would also be defined in a linear manner i.e. with respect to a default dimensional value of 1).
Therefore the deeper holistic reason why the Riemann zeta function remains undefined for s1 = 1, is due to the fact that at this value - and only this value - the dynamic interactive relationship as between distinctive Type 1 and Type 2 aspects of the number system is broken. For all other values (where s1 ≠ 1), a distinctive Type 2 formulation of the number system is possible.
Now this is utterly devastating, because it implies that the one interpretative model which - by definition - is unsuited to understanding the truly dynamic relative nature of the number system, is that which currently defines the universally accepted current mathematical approach!
In other words, given that the true nature of the number system is dynamic and relative, combining distinctive aspects of quantitative independence and qualitative interdependence respectively, then we cannot hope to coherently understand the nature of that system in terms of an approach that is merely quantitative in an absolute manner!
Just as there is one Type 1 extreme (where s1 = 1) where the relationship as between quantitative and qualitative aspects of the number system is broken, likewise there is one Type 2 extreme (where s2 = 1) where likewise this relationship is broken.
We define the Zeta 2 function in general terms as,
ζ(s2) = 1 + s21 + s22 + s23 + ...
Thus when s2 for example = 2, we get
ζ(2) = 1 + 21 + 22 + 23 + ...
Then in (linear) Type 1 terms, this ultimately would be given as,
11 + 21 + 41 + 81 + ...
However, in direct terms here, there is also an important (circular) Type 2 aspect.
So written in Type 2 terms,
ζ(2) = 10 + 11 + 12 + 13 + ...
Then combining both Type 1 and Type 2 aspects,
ζ(2) = 11 * 10 + 21 * 11 + 41 * 12 + 81 * 13 + ...
So now, the function is defined with respect to both (linear) quantitative and (circular) qualitative aspects (that dynamically interact in two-way fashion with each other).
However when s2 = 1, we have again - from the opposite perspective - a breakdown with respect to the relative nature of number.
For ζ(1) = 10 + 11 + 12 + 13 + ... is defined this time purely with respect to the Type 2 aspect of number (where no meaningful Type 1 aspect can be provided).
Therefore with respect to the Type 2 aspect of the number system (directly represented by the Zeta 2 function) the only value for which the relative interactive nature of number is broken is where
s2 = 1.
Now in general terms,
ζ(s2) = 1 + s21 + s22 + s23 + ... = 1/(1 – s2).
Therefore, when s2 = 1, ζ(s2) = 1/1 – 1) = ∞.
So we can see the complementary nature of the Type 1 and Type 2 aspects of the number system in evidence here.
Again the one value for which the Zeta 1 function is undefined is where s1, representing a dimensional value = 1.
And the one value for which the Zeta 2 function is undefined is where s2, representing a corresponding base value, = 1.
Once more, the key significance of these two values is that the Riemann Zeta function cannot be defined with respect to its qualitative holistic meaning (where s1 = 1) and then in turn cannot be defined with respect to its quantitative analytic meaning (where s2 = 1).
However for all other values of both s1 and s2, ζ(s1) the Riemann zeta function and ζ (s2), when coherently interpreted, represent in dynamic two-way relative fashion, the interaction of quantitative notions of number independence with qualitative notions of number interdependence respectively.