Yesterday we explored one way in which the Zeta 2 Function can be used to show an extremely interesting connection with the prime numbers.
So again by using the series,
y = 1 + s + s^2 + s^3 +....+ s^(n - 1) then setting s = 1, by a process of continual differentiation of y with respect to s, we can devise a simple rule to determine whether a number is prime (or not prime).
Ultimately (by differentiating down to the linear form of the original expression for y), we derive a numerical result = (n - 1)! + (n - 2)!
Again when n is prime this entails that this result will be divisible (for n > 3) by all prime factors from 2 to n inclusive (and only these primes). It will also always be divisible (again for n > 3) by the sum of all the natural numbers from 1 to n inclusive!
Now this result is especially interesting in that it provides an inversion of the usual means of testing for primeness!
Conventionally to test if a number is prime, we look for factors for that number and if no such (reduced) factors can be found, we conclude that the number is indeed prime. The problem is that with very large numbers a huge combination of potential factors exist which makes testing for primality very time consuming!
However in this case we head in the opposite direction whereby we test for primes through establishing whether the number in question is itself a factor of some larger number (that is determined by the precise procedures in question).
So in this case using the Zeta 2 expression, by establish that n is a factor of each of the derivative expressions of y (where s = 1), we conclude therefore that it is prime!
Now we have already met another example of this procedure with respect to the denominators of the Zeta 1 expression (where negative odd integral values of s are concerned).
So here we start with the value s (representing the dimensional power of the Zeta 1 Function) that is even.
Then through the Functional Equation a direct link is established with the corresponding dimensional value for the Zeta Function i.e. 1 - s.
And we then concluded that if the denominator of ζ(1 - s) is then divisible by s + 1 that s + 1 is thereby a prime number!
So for example when s = 10,
1 - s = - 9.
And the denominator (absolute value) of ζ(- 9) = 132.
Therefore if 132 is divisible by s + 1 (= 11), then s + 1 is prime.
And of course we can easily verify that that it is indeed true in this case!
However this would suggest that there are in fact deep close connections as between the Zeta 2 Function and the corresponding Zeta 1 Function (for negative values of s < 0).
Thus we cannot give a meaning in the standard cardinal sense to numerical results of the Riemann Zeta Function for values of s < 0 (indeed for values < 1).
However once we recognise that the corresponding ordinal notion of number is qualitatively distinct (relating directly to the Zeta 2 Function), then indirectly we can then interpret numerical values for the Riemann Function (s < 1) in a meaningful fashion.
To appreciate this properly we need to move to a dynamic relative notion of number. The cardinal aspect can then be identified with the standard quantitative interpretation (where number is now understood as independent in a relative manner).
The ordinal aspect relates to the corresponding relational aspect of number as interdependent (which constitutes the unrecognised qualitative aspect of interpretation).
In this context the trivial zeros of the Zeta Function have a pure ordinal meaning representing perfect interdependence (so that any distinctive cardinal aspect no longer remains).
So when 0 is used to represent the value of the Riemann Zeta Function for all negative even values of s, it relates to a pure ordinal meaning that is of a directly qualitative nature.
Now the values of the Riemann Zeta Function for all the negative odd values of s are more problematic in that distinct rational values arise. However here the denominators - rather than simply cardinal values - represent number relationships that have both ordinal and cardinal aspects.
In other words, pure cardinal meaning (as in conventional linear mathematical interpretation) and pure ordinal meaning (as in pure circular mathematical interpretation) represent two extremes.
And once again we have seen that the pure circular aspect arises for all even numbered roots (and corresponding dimensions) as perfect complementarity can be maintained as between positive and negative roots).
So when we look at the 4 roots of unity, 1 has a complement in - 1; likewise i has a complement in - i. Thus a pure circular interdependence exists with respect to these 4 roots. In corresponding qualitative terms, pure circular interdependence applies to 4 as a dimensional means of mathematical interpretation! (This for example helps to explain, with respect to Jungian Psychology, why mandalas based on the geometrical structure of the four roots of unit can operate as powerful symbols of integration)!
Now once again the positive dimensional number implies the (indirect) rational understanding of interdependence (as the complementarity of opposite poles).
The negative dimensional number implies the direct intuitive recognition of such interdependence (which is nothing in phenomenal terms). So this is the true qualitative explanation of why all the trivial zeros are associated with negative even dimensional numbers for the Zeta Function!
However we do not have perfect complementarity with odd integer roots! Likewise we do not have perfect complementarity with the odd integer dimensions. Thus the intuitive attempt to grasp the nature of holistic interdependence with respect to the interpretation of the Riemann Zeta Function with respect to the negative odd integers (as dimensional values of s) necessarily implies also a rational aspect!
So in a future entry, I will look in detail at the significance of the numerical values for the negative odd integers of s (showing precisely how they are related to the Zeta 2 Function and how they combine both ordinal and cardinal elements with respect to interpretation.