A very puzzling feature exists as to the nature of results for the Riemann zeta function where s is a positive integer.
Now apart from the case where s = 1, the function has a finite value However the form of this value differs distinctly as between even and odd values.
In the case where s = 2, ζ(2) = π2/6 and in fact where s is a positive even integer, the result can always be expressed in the form πs/k, where k is a rational number.
So for example when s = 4, ζ(4) = π4/90; when s = 6, ζ(6) = π6/945; when s = 8,
ζ(8) = π8/9450; when s = 10, ζ(10) = π10/93555. Now up to s = 10, k is an integer. However this pattern is broken when s = 12, with ζ(12) = 691π12/638512875.
The denominators associated with all these values are especially interesting in their own right for the following reasons.
When s is a power of 2, the denominator of ζ(s) is divisible by all the primes (and only those primes) to s + 1.
When s is not a power of 2, the denominator of ζ(s) is divisible by all the primes except 2 (and only those primes) to s + 1.
For example the denominator - as we have seen - for ζ(12) is 638512875. Now 12 is not a power of 2. Therefore 638512875 is divisible by all the primes from 3 to 13 inclusive (and only those primes).
In fact 638512875 = 188.8.131.52.13.
Some years ago, I made an extensive study of these denominators (based on the data at my disposal) and this appears very much to be a pattern that universally holds for all denominators of ζ(s) where s is even.
However, where s is a positive odd integer, no such relationship involving powers of π (multiplied by a rational number exists). Indeed, no other convenient closed form relationship likewise appears to exist.
So therefore a fundamental difference characterises the behaviour of ζ(s) for positive even and odd integers respectively.
Now we can perhaps throw some holistic light now on the reason for this situation.
In the last blog entry I showed how the probability that 2 numbers chosen at random possess no common factors, could be expressed in terms of the Type 2 aspect of the number system as where the fractional exponent (expressing an appropriate root of unity) consists of the 2 numbers involved.
So again in the example given there, if 2 numbers (chosen at random between 1 and 100) are 16 and 60, then this could be expressed in the Type 2 system as 116/60.
So in this case the fact that the two numbers 16 and 60 have common factors (i.e. proper factors), can be expressed in Type 2 terms by the fact that both the numerator and denominator of the fraction (expressing a root of 1) likewise contains a common factor.
And again as we have seen, the resulting root (– .1045 +.9945i) provides an indirect quantitative means of expressing the notion of the 16th member (of a group of 60). So the very point about this is that what we have here the harmonious reconciliation of both the ordinal and cardinal notion of number. And this entails the corresponding reconciliation of both circular and linear number notions. So the 16th root in question here represent a point on the unit circle that is connected from the centre of the circle by a unit line! So the 16th root (in ordinal terms) represents one of the 60 (cardinal) roots of 1.
Likewise the proper understanding of this relationship (entailing both quantitative and qualitative notions of number) entails a corresponding reconciliation of holistic (intuitive) and analytic (rational) understanding (that are circular and linear with respect to each other).
However if we now consider the probability of 3 numbers chosen at random containing no common factor, this cannot be expressed with respect to the Type 2 system. Whereas any two numbers could be used as a fractional index (of 1) representing one of the roots of 1, the remaining number could not be expressed in this manner.
So for example our 3 numbers chosen at random (again between 1 and 100) are 12, 25 and 71
we could express any two together as a fractional index (of 1) e.g. 12/25, 12/71 or 25/71. However in each case there will be one number left over (that does not represent a fractional root),
Therefore the pure circular/linear relationship that characterises each root (in both quantitative and qualitative terms) cannot therefore apply in this case.
Indeed long before any mathematical considerations, I had looked deeply into this issue from a pure holistic perspective (relating to "higher" contemplative stages of development).
Now normal development is characterised by a considerable degree of linear (1-dimensional) understanding that allows for unambiguous dualistic distinctions with respect to phenomena.
So the first major stage of contemplative spiritual development culminates in 2-dimensional appreciation, where opposite external and internal polarities (which fundamentally condition all phenomenal understanding) are understood in nondual fashion as ultimately fully complementary with each other. Expressed in holistic mathematical terms, these two polarities are understood paradoxically as + 1 and – 1 with respect to each other (so that the signs can interchange) when interpreted in the conventional linear manner.
Now of course these same polarities necessarily condition all mathematical understanding and this is why the conclusions that apply in contemplative spiritual terms equally apply in a holistic mathematical context!
I then spent many years trying to come up with a neat holistic mathematical manner of expressing the next 3-dimensional stage before eventually realising that no such way existed.
So I gradually came to accept that the even dimensional numbers represented the more passive stages, where a certain contemplative equilibrium (entailing the balanced use of both reason and intuition) prevailed. However the odd dimensional stages represented more active stages, where the previous equilibrium would temporaily break down, requiring a new period of conflict btween reason and intuition before a "higher" contemplative equilibrium could once again be restored.
In particular during the 3rd dimensional stage, I was aware of a marked discrepancy as between new holistic insights (formed during the previous contemplative stage) and my ability to manage practical duties at the conventional level (requiring linear reason).
So the key problem therefore during the 3rd stage was this mismatch as between (circular) intuition and (linear) reason. This was then eventually resolved to a degree at the 4th dimensional stage, where a much more refined experience of the interaction between fundamental poles took place.
So in holistic terms at the 4th dimensional stage, balance is once more restored as between external and internal polarities (as - relatively - positive and negative with respect to each other) and the crucial whole and part polarities (i.e. qualitative and quantitative), that are now understood in refined holistic mathematical terms as real and imaginary with respect to each other.
So though we are speaking here about the four roots of unity, they acquire in this development context an entirely distinctive holistic meaning! And this is potentially true with respect to every mathematical symbol and relationship!
So when we now look at the Riemann zeta function, we can see the same mismatch with respect to values of s representing the odd integers.
Whereas linear and circular aspects (in both quantitative and qualitative terms) are successfully integrated with each other (where s is even) a certain displacement is in evidence (where s is odd). So the rational numbers are transferred to the LHS of the function (associated with negative values for s) whereas the irrational aspect remains on the RHS.
So the values of the Riemann zeta function (where s are positive odd integers) are indeed irrational but not in a perfect manner as represented by the transcendental number π. So no displacement of the rational aspect takes place for even values of s, which is indicated by the fact that the value of ζ(s) for all negative even integer values of s = 0.
There is yet another simple way of appreciating the difference as between the value of ζ(s) for positive even and odd integers respectively.
When one obtains an even number of roots of 1, a direct complementarity is in evidence where each of one half of the roots can be exactly matched in each case by their negative counterpart (in the other half).
However where an odd number of roots is concerned, such complementarity does not exist, with one of the roots (i.e. 1) standing somewhat apart, while the remaining roots exist as complex conjugates of each other.
Now if we briefly look at the probability that 4 numbers chosen at random will not contain a common (proper) factor, this will be given as 1/ζ(4) = 90/π4 = .9239...
We can therefore formulate this in Type 2 terms, this time requiring fractional numbers for 2 roots.
So if for example (again choosing at random from 1 - 100) we get for example the four numbers, 7, 31, 43 and 78, we could express these in Type 2 terms as 17/31 and 143/78. So the issue now depends on whether both numerator and denominator of both 7/31 and 43/78 contain a common factor.
Alternatively expressed this implies the probability that the product of the four numbers (i.e. 7 * 31 * 43 * 78) is not divisible by a prime factor (raised to the power of 4).