I will attempt to develop further the insights in yesterday's blog entry that were directly related to the remarkable formula for 1/24 (given by Ramanujan in his 1st letter to Hardy).
Once again we showed - using this formula - how a fascinating sum over the natural number expression can be given for – 1/12.
Now according to the Riemann Zeta function,
ζ(– 1) = – 1/12. Likewise, perhaps surprisingly, ζ(– 13) = – 1/12.
So if we let 1/s = – 1/12, then s = – 12. Therefore ζ(– 13) = ζ(s – 1).
Then we have the remarkable result that ζ(s – 1) = 1/s
Therefore the result of the Riemann zeta function for ζ(– 13), where s = – 12 is given by the reciprocal of s, = – 1/12.
In other words, the result of the function for this negative value of s, where s relates to the dimensional power (to which successive natural numbers are raised) is given directly as the reciprocal of that same dimensional number.
So if we were to write out - according to the Riemann zeta function - the numerical expression corresponding to ζ(– 13), we would obtain the following,
ζ(– 13) = 113 + 213 + 313 + 413 +... = – 1/12.
However clearly this result has no meaning in terms of the standard analytic manner of interpreting the sums of series.
The reason for this is that a new holistic manner of interpretation is now required to make sense of this result (which is non-intuitive in conventional mathematical terms).
And there is a deep clue in the very nature of the result as to what is involved.
In conventional analytic terms, the numerical sum of terms of an expression is treated in a reduced quantitative fashion, where ultimately it is interpreted with respect to the base values of each term (raised to the default dimensional power of 1).
Thus 113 from this perspective = 1 (i.e. 11), 213 = 8092 (i.e. 80921), 313 = 1594323 (i.e. 15943231) and so on.
Thus when we attempt to add these terms, the series quickly diverges (to what in reduced quantitative terms is expressed as ∞).
However when we invert this whole process and now look at the series directly from the dimensional power, with no regard for the specific base values that arise, we can see that the actual result i.e. – 1/12 can in fact be meaningfully related to the common dimensional - rather than varying base - numbers that arise.
In other words, the true meaning of the series is holistic where the overall dimensional structure of terms - rather than the specific value of each term - is what is relevant in this context.
So from this holistic perspective, we cannot give the terms in the series expansion a separate i.e. part, identity, as each term only has meaning in the overall relational context of the series which is directly of a qualitative holistic nature, which however can then - indirectly - be given a distinct quantitative interpretation!.
Therefore, once again,
ζ(– 13) = 113 + 213 + 313 + 413 +... = – 1/12, has no meaning from the standard analytic perspective.
However we have already provided an alternative expression which does - indirectly - offer an intuitively satisfying value for – 1/12.
And remarkably when we look at this alternative expression the numerators of each term, can be seen to exactly replicate the corresponding terms in the Riemann expansion for ζ(– 13).
So, as given in yesterday's entry,
2. 113/(1 – e2π ) + 2. 213/(1 – e4π ) + 2. 313/(1 – e6π ) +... = – 1/12.
Now, to achieve direct comparison with the Riemann function we can take the 2 out of the numerator and divide the denominator in each case by 2.
So the numerator of the 1st term for example would now be 113, bearing direct comparison with the corresponding term in the Riemann zeta function.
The denominator of this term would then be given as (1 – e2π )/2.
Therefore the significance of the denominator term in each case is that it provides a ready means of providing the necessary holistic conversion for the corresponding terms in the Riemann zeta function.
And just as analytic interpretation is strongly based on linear rational notions - where literally all real quantitative values are ultimately interpreted in 1-dimensional terms (as lying on the number line) - holistic interpretation, indirectly, is strongly based on circular quantitative notions (relating to the unit circle in the complex plane) that directly are understood in a intuitive manner.
So we can readily see in our denominator "conversions" this strong circular aspect, with the dimensional value to which e is raised representing natural number multiples of 2π (as the circumference of the unit circle).