I have often been intrigued as to the contents of that famous 1st letter of Ramanujan to Hardy. (I am indebted here to the supporting link provided on the excellent "You Tube" video "Ramanujan: Making sense of 1+2+3+... = -1/12 and Co.")
In connection with the Riemann zeta function, considerable mention has been made of the inclusion of this - apparently - nonsensical - formula for the sum of the natural numbers, i.e.
1 + 2 + 3 + 4 + ... = – 1/12.
However this seems a somewhat less eccentric result when one interprets it more formally as the value of the Riemann zeta function (for s = – 1).
In fact in the same letter, Ramanujan also claims that,
13 + 23 + 33 + 43 + ... = 1/120 (which corresponds in turn to the value of the Riemann zeta function (for s = – 3).
However though undoubtedly impressive, the derivation of these results is not in fact difficult (when one abandons the need to derive numerical results that conform to our normal expectations regarding number behaviour).
The first result can be derived easily with nothing less than elementary arithmetic and some creative juggling with infinite series.
Though the second result is more difficult to derive, it can be achieved through the same kind of series juggling (with a little help from elementary calculus).
So in a way I do not find it surprising at all, that Ramanujan - given his enormous intuitive gift for discovering original number patterns - would have come up with these results.
However some of his other results just take my breath away.
For example, he gives a formula for 1/24 that leaves me lost in admiration for its bold ingenuity. And we must remember that Ramanujan lived long before the age of calculators and computers that can greatly assist the testing of numerical patterns!
Now 24 is an especially interesting number. In fact it can be directly connected with that 1st result we mentioned (i.e. – 1/12 as the sum of the natural integers)
It has been shown to have mysterious connections to the bosonic superstring theory in physics. It also has strong connections with the "Monster Group" which is the most symmetrical object yet discovered.
It also has direct connections to the "Leech Lattice" enabling the most efficient packing of objects to take place in a hypersphere of 24 dimensions.
Intriguingly the sum of the squares of the 1st 24 natural numbers = the square of an another natural number (i.e. 70) and this is the only case where the sum of squares of the natural numbers (up to n) leads to the square of another natural number!
Also from my own research, I found "24" especially important with respect to deriving a comprehensive holistic mathematical model of the basic personality types (with complementary connections, I believe to the basic "impersonality types" represented by bosonic string theory).
So in this context I was especially interested in seeing Ramanujan's unique formula for 24 (or rather 1/24), which is given as,
113/(e2π – 1) + 213/(e4π – 1) + 313/(e6π – 1) + ... = 1/24.
As this initially, seemed almost "too good to be true", I suspected that it was perhaps designed as an approximation. However this is not the case, with the value converging quickly to 1/24 after just a few terms.
Indeed, when the 1st 8 terms are added, the result = 1/24.0000000004...
It struck me in looking at this result that the formula can be provided in a slightly different - though intriguing - fashion.
e2π = 1– i .
Therefore we can rewrite Ramanujan's formula as,
113/(1– i – 1) + 213/(1– 2i – 1) + 313/(1– 3i – 1) + ... = 1/24.
Then because 1/24 is so closely connected with – 1/12, we can therefore now provide an interesting sum over the natural numbers expression for – 1/12, i.e.
2. 113/(1 – 1– i ) + 2. 213/(1 – 1– 2i ) + 2. 313/(1 – 1– 3i ) +... = – 1/12 or alternatively,
2. 113/(1 – e2π ) + 2. 213/(1 – e4π ) + 2. 313/(1 – e6π ) +... = – 1/12
There is a fascinating connection here with earlier work I concluded on this blog site, where I attempted to approximate values for Riemann zeta function over different ranges (for positive and negative values of s).
In particular in "Approximating ζ(s) between 0 and – 1" , I noticed a remarkably close connection as between the value of ζ(– .5) and the corresponding value of e– π/2 (i.e. i i).
I then used an approach based on the value of e– π/2 to approximate results for all values of s in this range between 0 and – 1.
So the approximation there for ζ(– 1) = – 1/12 was given as – 2e– π = – 2i2i.
The simple approximation therefore here for 1/24 = e– π, i.e. i2i. So this result for 1/24 - though in itself not terribly accurate - indicates however a deep connection with ekπ expressions (where k is an even integer).
This would suggest strongly to me therefore that there is an intimate connection in terms of the "non-intuitive" values corresponding to negative values of s for the Riemann zeta function and corresponding sums over natural numbers expressions that involve ekπ.
So this provides some perspective on the "Alice in Wonderland" nature of behaviour for values of s > 1 and corresponding values of s < 0 respectively.
As I have repeatedly stated on these blogs our conventional notion of addition - where results intuitively conform to our expectations - is in fact based on a reduced interpretation of number behaviour where the qualitative (holistic) aspect is reduced in a quantitative (analytic) fashion.
In particular this involves treating the whole (in any context) as merely the reduced expression of its constituent (independent) parts.
Now this approach does indeed work - at least in quantitative terms - for all values of the Riemann zeta function (where s > 1).
However it then breaks down badly for all corresponding values of the function (where s < 0).
Here, infinite series (in the sums over natural numbers expressions) though clearly divergent (from the conventional analytic perspective) yet result in definite finite values.
Now the clue here is that number behaviour in fact switches from a linear (analytic) to a - relative - circular (holistic) pattern. Putting it in more psychological terms, we have a switch here in number behaviour from what accords with conscious (analytic) interpretation to a new behaviour which now accords directly with unconscious (holistic) appreciation.
The very nature of the conventional analytic approach is that it is - literally - 1-dimensional, in that all real results are treated as numbers lying in the same number line (with dimension s = 1).
And this linear approach works (for values of s > 1) where intuitively satisfying quantitative results emerge for the Riemann function.
However when we apply this linear approach to values of the function (where s < 0), there can be no intuitive resonance with these results.
What in fact has happened is that dimensional behaviour has now directly to a circular notion (as indicated by a power of e = kπ).
Thus when we translate the "non-intuitive" sum over natural numbers expressions (as previously interpreted in linear dimensional terms) to a new sum over natural numbers expressions (as now interpreted in circular dimensional terms) an intuitively satisfying quantitative result can emerge.
So by using Ramanujan's ingenious formula for 1/24, I have shown here how the value of ζ(– 1) = – 1/12 can be equally expressed in a circular dimensional fashion as an infinite sum over natural numbers, which converges quickly towards the expected answer.
This would suggest that in principle that this circular type "conversion" is equally possible for all values of the Riemann function (where s < 0).
It would also suggest that associated with these new sum over the natural numbers expressions are corresponding "circular" product over the primes expressions.
And it is through such "conversions" that the - at present - hidden world of the Riemann function for values of s < 0 can become intuitively accessible in quantitative terms.
However this implies the much deeper realisation that the number system itself must be understood in a dynamic interactive manner, entailing both complementary quantitative (analytic) and qualitative (holistic) expressions. In psychological terms this requires the corresponding full integration of both conscious and unconscious aspects of mathematical understanding.