We are continuing here our explorations from the previous two blog entries.
Once again we have seen with respect to the value of ζ(– 13) = – 1/12, a remarkable connection as between the analytic (linear) interpretation of the Riemann zeta function and a "new" holistic (circular) interpretation, based on Ramanujan's ingenious formula for 1/24.
And whereas the linear interpretation leads to a seemingly "nonsensical" result that is completely non-intuitive from this perspective, the corresponding circular interpretation leads by contrast to a fully intuitive result (as the indirect quantitative expression of an interpretation that is directly of a holistic intuitive nature).
And the numerical answer here is directly related in a very simple way to the corresponding value of s (representing the common power or exponent to which all the natural numbers are raised).
So whereas in standard interpretation, the value of a series the sum of terms is calculated with respect to base numbers, where each term is viewed as making a separate independent contribution, by contrast with holistic interpretation, the value of a series of terms is viewed in terms of the overall interdependent structure of the series, now calculated indirectly in quantitative terms with respect to dimensional numbers.
So once again with respect to ζ(– 13) = 113 + 213 + 313 + 413 + ..., in standard interpretation, we calculate each individual term separately with respect to its base value (raised to the default dimension of 1).
Thus 113 = 11, 213 = 80921, 313 = 15943231, 413 = 671088641 and so on.
We the attempt to aggregate these values through addition of each term, whereby it quickly becomes apparent that their sum diverges to ∞ (from this perspective).
However, as the correct value of ζ(– 13) = – 1/12, this clearly shows that the standard (linear) interpretation does not apply in this case.
By contrast with holistic interpretation, we treat the overall series as a collective unit (where no separate meaning attaches to each individual term). Rather, we are now viewing the series with respect to a common structural feature of shared interdependence by all the terms, which is given through the dimensional value (to which each term is raised).
Now again, strictly according to the Riemann zeta function, the dimension here is – 13.
So ζ(– 13) = 1/1 – 13 + 1/2 – 13 + 1/3 – 13 + 1/4 – 13 + ... = 113 + 213 + 313 + 413 + ...
And with s = – 12, – 13 = s – 1.
And we can see that the value of the series i.e. – 1/12 = 1/s.
However to indirectly express this shared feature of interdependence with respect to the collective series in a quantitative manner, we need to apply a holistic "conversion" with respect to each term, through division by an appropriate "circular" component.
So the 1st term, i.e. 113, is thereby divided by (1 – e2π)/2, the 2nd term, i.e. 213 by (1 – e4π)/2, the 3rd term, i.e. 313 by (1 – e6π)/2, the 4th term, i.e. 413 by (1 – e8π)/2 and so on.
We are then enabled to add the numerical value of each separate term in the accepted quantitative manner to obtain a value for the infinite series of terms = – 1/12.
Therefore the vital point to grasp here is that the value of ζ(– 13) with respect to the Riemann zeta function, properly conforms to a holistic rather than analytic interpretation of number. Now properly this holistic interpretation is directly of an intuitive qualitative nature, by which the overall interdependence of terms in the series is appreciated. However indirectly - as we have seen - it can then be given an indirect quantitative value, through applying each term to an appropriate "circular" number conversion.
However the deeper point here is that in order to be meaningful, all values of the Riemann zeta function (for s < 0) must be given such holistic type interpretation. Once again, though in direct terms such appreciation is of a direct qualitative intuitive nature, indirectly it can be given a coherent quantitative expression.
Only then, can the true significance of Riemann's functional equation be properly understood, i.e. where the function for positive values of s > 1 is paired with the same function for corresponding values of 1 – s (which are negative).
In other words the functional equation to be properly interpreted must be understood in a dynamic relative manner. Thus what has an analytic (quantitative) interpretation where s > 1, has a corresponding holistic (qualitative) interpretation where (1 – s) < 0 and vice versa.
So quantitative notions of number independence can have no strict meaning in the absence of corresponding qualitative notions of number interdependence (and vice versa). Thus number independence and number interdependence are strictly relative notions, which mutually imply each other in a dynamic interactive fashion.
Though the rational values for other negative (odd) integer values of s with respect to the Riemann zeta function are more difficult to fully unravel, the basic points that I have made here with respect to ζ(– 13), will hold in all cases.
In fact the number 12 plays a special role with respect to all these values.
The denominator of the value of the Riemann zeta function (for all odd integer values) is divisible by 12.
So the denominator of ζ(– 1) = 12 (which is divisible by 12).
The denominator of ζ(– 3) = 120 (which is divisible by 12).
The denominator of ζ(– 5) = 252 (which is divisible by 12).
The denominator of ζ(– 7) = 240 (which is divisible by 12).
The denominator of ζ(– 9) = 132 (which is divisible by 12).
The denominator of ζ(– 11) = 32760 (which is divisible by 12).
The denominator of ζ(– 13) = 12 (which is divisible by 12).
There is another important observation that can be made here.
The behaviour of these denominators is itself strongly subject to cycles of 12.
So as we know ζ(– 1) = – 1/12.
Then when we move on through a complete (negative) cycle of 12, ζ(– 13) = – 1/12.
Though the numerator from here on steadily grows larger and larger, the denominator still remains strongly subject to such cyclical behaviour.
Thus the denominator of ζ(– 25) = 12 and the denominator of ζ(– 37) = 12.
Now the denominator of ζ(– 49) breaks this trend = 132.
However it is once again restored for the vast majority of further cycles of 12.
In particular for ζ(– 121), where s = 120, the denominator = 12.
This would suggest that all these values can be readily expressed with respect to the "central" value for ζ(– 13).
Therefore once again for example, ζ(– 3) = 1/120.
The appropriate "converted" holistic sum of terms for this series is then given by,
113/5(e2π – 1) + 213/5(e4π – 1) + 313/5(e6π – 1) + .... = 1/120 or alternatively,
113/5(1– i – 1) + 213/5(1– 2i – 1) + 313/5(1– 3i – 1) + ... = 1/120.