Ramanujan's Letter (5)

I would like to move on to another fascinating result from Ramanujan's 1st letter to Hardy.

Ramanujan asks us to take all numbers with an odd number of prime divisors. This had me initially scratching my head as to what this might entail! He then gives a list of these numbers up to 50, i.e..

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47

It is noteworthy that - apart from 30 and 42 - these provide a list of the prime nos. to 50.

So in all but 2 cases up to 50 an odd number of prime divisors implies just one divisor, i.e. the prime number itself.

The 2 exceptions here, 30 and 42 both contain 3 dissimilar prime divisors i.e. 30 = 2 * 3 * 5 and 42 = 2 * 3 * 7.

He then gives a formula for the sum of the squares of the reciprocals of these numbers,

i.e. 1/2² + 1/3² + 1/5² + ... + 1/30² + 1/31² + ...  = 9/2π².

Now this bears direct comparison with the Riemann zeta function (for  s = 2), where, the sum of the squares of the reciprocals of the natural numbers, i.e.

1/1² + 1/2² + 1/3² + 1/4² +  ...  = π²/6.

What is remarkable here, is that if one attempts to sum up the reciprocals of the prime numbers that it will converge very close to the value 9/2π² as all the terms up to 1/29² relate to primes.

So one could be easily tempted therefore to draw the wrong conclusion that the sum of the squares of reciprocals of the prime numbers converges to a neat expression involving the corresponding square of π.

However what is very interesting is that - whereas with the sums of squares of the natural numbers - the square of π relates to the numerator of the result, in the case of the reciprocals of the squares of the primes the corresponding square of π relates to the denominator.

However it is still noteworthy that the sum of squares of the reciprocals of primes approximates so closely to a square of π expression.  


This pattern is even more in evidence in relation to the reciprocals of the same numbers (i.e. with an odd no. of prime divisors) when raised to the power of 4.

Ramanujan gives the following neat formula (showing definite similarities with the previous formula
for reciprocals raised to the power of 2).
 
1/2⁴ + 1/3⁴ + 1/5⁴ + ... + 1/30⁴ + 1/31⁴ + ... = 15/2π⁴.

The potential prime number trap is even more closely in evidence here.

For example if one adds up the reciprocals (raised to the power of 4) of all the primes to 100, one obtains the result 14.999649.../2π⁴. One might then be forgiven for assuming that with the addition of further prime terms that the numerator of the expression here would converge eventually to 15.

However the true result of adding the reciprocals (raised to the power of 4) of all numbers divisible by an odd number of prime factors up to 100 = 14.999976.../2π⁴.

So sometimes initial appearances may prove deceptive. However once again it is indeed striking that the sum of the reciprocals of the primes (raised to the power of 4) approximates so closely to 15/2π⁴.


Then there are further surprises arising. One might again be tempted to think that with nice expression in powers of π arising where the reciprocals of relevant terms are raised to the power of 2 and power of 4 respectively, that this pattern would be repeated for all powers (entailing positive even integers).

However this does not appear to be the case. Certainly from my own investigations, there was no evidence of any simple number pattern (relating to powers of π) emerging with other even integer dimensions.

Likewise I could find no simple pattern entailing reciprocals of numbers with an even number of prime factors.

So again it is so ingenious how Ramanujan made such discoveries.

Incidentally, he also gives a simple formula for the calculation of the frequency of numbers with an odd number of prime factors.

This is given as 3n/π².

This would imply that where n = 100, we should expect  30.396...  i.e. 30 (rounded to the nearest integer). And in fact this is exactly the number we obtain containing the 25 primes up to 100 and 30, 42, 66, 70 and 78.  

Of course most of the numbers here to 100 are prime. However as we move higher up the number scale, the relative percentage of prime to all divisors (comprising an odd number of prime factors) would steadily decrease.

Ramanujan's Letter (4)

We ended yesterday's entry with the "converted" holistic expression for ζ(– 3), i.e.

1¹³/5(e^2π – 1) + 2¹³/5(e^4π – 1) + 3¹³/5(e^6π – 1) + ...   = 1/120.

Perhaps this is better written as,

2.1¹³/10(e^2π – 1) + 2.2¹³/10(e^4π – 1) + 2.3¹³/10(e^6π – 1) + ...  = 1/120.   

And then the corresponding "converted" holistic expression for  ζ(– 5),

= 2.1¹³/21(1 –  e^2π) + 2.2¹³/21(1 –  e^4π) + 2.3¹³/21(1 –  e^6π) + ....  = – 1/252. 

Intriguingly, the "converted" holistic expression for ζ(– 1) is exactly the same as that for ζ(– 13)

=  – 1/12.

In this respect, it is just like a clock that at both at 1.00 hours and at 13.00 hours will show the same time (representing 1 AM and 1 PM respectively). 

Though the early results for the smaller negative odd integer values of ζ(1 – s) involve the simple reciprocal of a whole number, later results look much more unwieldy with the numerator now attaining an increasingly large size. as can be seen here.

However in principle they can all be equally expressed in terms of the standard holistic "conversion" for ζ(– 13).

So ζ(– 11) =  2.691.1¹³/2730(e^2π – 1) + 2.691.2¹³/2730(e^4π – 1) + 2.691.3¹³/2730(e^6π – 1) + ...  

= 691/32760.

I now wish to comment on the true qualitative holistic significance of these number transformations.

So again from the standard linear perspective the series for ζ(– 1) seems utterly straightforward, representing the familiar sum of the natural numbers i.e. 1 + 2 + 3 + 4 + ... 

However, though we expect the  result of this infinite series to diverge - in conventional terms - to  

∞, in fact according to the Riemann zeta function, the result = – 1/12.

The implications here could not be more significant, for when one properly understands the true reason for this "strange" result it then becomes readily apparent that our present interpretation of number operations (such as addition) is simply not fit for purpose.

In the conventional approach both the quantitative and qualitative aspects of number understanding are formally abstracted from each other leaving the misleading impression that number operations can be understood in an absolute quantitative manner (without the need for any qualitative considerations).

So, again in conventional terms, the natural numbers are treated in an absolute independent manner (with respect merely to their quantitative identity).

However this position is strictly quite untenable, for if numbers were truly independent of each other in an absolute fashion, then it would not be possible to understand them in relation to each other.

So this all important aspect of number interdependence - which inherently is of a qualitative nature - in conventional mathematical terms is simply reduced in a quantitative manner.

And this reduced quantitative way of thinking leads directly to a reduced notion of the whole as merely the sum of its constituent parts.

Thus, from this perspective, when for example we attempt to sum an infinite series, we do not give this series a proper collective identity (as befits its infinite status) but rather merely a part identity as the aggregate of its individual finite elements.

And we are so attuned to reducing whole to part notions in conventional mathematical terms that we no longer even question this approach.

However to coherently interpret the Riemann zeta function, we need to radically change the prevailing mathematical orthodoxy.

For rather than number misleadingly being treated with respect to an absolute quantitative identity, number must now be understood in truly relative terms (with explicitly recognised quantitative and qualitative aspects).

In recent weeks, I have been showing how a hidden qualitative aspect must be introduced to coherently interpret values of the Riemann zeta function for values of s > 1.

So as well as the recognised linear element, an unrecognised circular aspect is likewise involved.

However because intuitively meaningful quantitative results emerge for the function (within this range of s) we misread the dynamic relationship between both linear and circular aspects of the number system that is properly involved, thereby reducing interpretation in a linear quantitative manner.

However a mirror image picture then emerges with respect to values of the function for s < 0.

Again, both quantitative and qualitative aspects are involved. However, here the numerical result that is given primarily represents a true whole (i.e. qualitative) interpretation of the series (which is then indirectly converted in a quantitative manner).

Thus because of the complete lack of a recognised holistic aspect in conventional mathematical terms, the results that arise must necessarily remain non-intuitive from a limited linear perspective.

So in the most beautiful manner possible the Riemann zeta function - when correctly understood - shows how both quantitative (analytic) and qualitative (holistic) aspects fundamentally interact in a dynamic two-way relative fashion throughout the number system.  

Ramanujan's Letter (3)

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 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 then 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 –  e^2π)/2, the 2nd term, i.e. 2¹³ by (1 –  e^4π)/2, the 3rd term, i.e. 3¹³ by (1 –  e^6π)/2, the 4th term, i.e. 4¹³ by (1 –  e^8π)/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, 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(e^2π – 1) + 2¹³/5(e^4π – 1) + 3¹³/5(e^6π – 1) + ....  = 1/120  or alternatively,

1¹³/5(1^{– i}  – 1) + 2¹³/5(1^{– 2i}  – 1) + 3¹³/5(1^{– 3i}  – 1) + ...   = 1/120.

Ramanujan's Letter (2)

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 –  e^2π ) + 2. 2¹³/(1 –  e^4π ) + 2. 3¹³/(1 –  e^6π ) +...    = – 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 –  e^2π )/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).  

Ramanujan's Letter (1)

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 = 70² 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/(e^2π – 1) + 2¹³/(e^4π – 1) + 3¹³/(e^6π – 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.

e^2π  = 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 –  e^2π ) + 2. 2¹³/(1 –  e^4π ) + 2. 3¹³/(1 –  e^6π ) +...    = – 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 e^kπ  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 e^kπ.

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 interpreted 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 on 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 switched 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.  

Twin Primes (2)

When one accepts that the sum of twin primes (apart from the first pairing 3 and 5) is divisible by 12, this then implies that the product of these twin primes when divided by 12 will always leave a remainder of 1.

This is due tho the fact that (n + 1)(n – 1) = n² – 1. And if the sum of twin primes = 2n is divisible by 12, (i.e. n is divisble by 6), then n² is thereby divisible by 12; so when n² – 1 is divided by 12, it leaves a remainder of 1.

Expressed in an alternative fashion the result = k – 1/12 (where k is an integer).

This suggests a possible connection with the result of the Riemann zeta function when s = – 1,

i.e. ζ( – 1) = – 1/12.

In fact, in this context it is interesting to note that the denominators of the Riemann zeta function for all negative odd integers (– 1, – 3, – 5,......) - as with the sum of twin primes - appear to be always divisible by 12!        
 
Thus if n – 1 and n + 1 represent twin primes (other than 3 and 5) then n² is divisible by 12.

In fact it always appears to be divisible by 36!

When we calculate the result of  n²/36 with the value of  n here relating to  the average of successive twin primes, an interesting pattern results.

For 5 and 7,           n²/36 = 36/36      = 1 

For 11 and 13,       n²/36 = 144/36    = 4

For 17 and 19,       n²/36 = 324/36    = 9

Now if we watch the series on the right the terms seem to be increasing by 1, 3, 5,... Alternatively these results represent the successive squares of the natural numbers.

For 23 and 25,       n²/36 = 576/36 = 16 (though of course 25 is not prime). Interestingly 25 also showed up as a "bogus prime" in yesterday's exercise. Then with 29 and 31 normal service is resumed.

For 29 and 31,      n²/36 = 900/36  = 25

Again we would expect that the next result on the right should be 36 and this occurs

For 35 and 37,     n²/36 = 1296/36 = 36 (even though 35 is not a prime).

With next pair,  normal service is again resumed.

For 41 and 43,   n²/36 = 1764/36  = 49

Now the next 2 squares of natural numbers are 64, 81 And these occur:

For 47 and 49,  n²/36 = 2304/36 = 64 (though 49 is not prime),

For 53 and 55,  n²/36 = 2916/36 = 81 (though 55 is not prime).

For 59 and 61  n²/36 =  3600/36 = 100.

What is remarkable here, just as in yesterday's exercise that the twin primes concur with the squares of 2, 3, 5 and 7 (for the first four pairs (after 5 and 7).

And even in all the other cases where the result is a square of a natural number that is not prime, one of the pairings is always prime (certainly up to 100).

This would suggest that a great way of searching for twin prime pairings would be to keep testing for
n²/36 = k² (where k = 1, 2, 3, ....).

Incidentally, it is again very interesting in view of the earlier observation that  ζ( – 1) = – 1/12, that the sum of all our results for k² (where k = 1, 2, 3, .....) = ζ( – 2).

Twin Primes (1)

I have mentioned before how the sum of all twin primes (excepting the first pair of 3 and 5) appears to be divisible by 12.

I then decided to look at the pattern of results obtained when the sum of successive pairs is divided by 12 and found an unexpectedly interesting pattern.

So when we divide the sum of 5 + 7 by 12, we obtain 1.

Then for the next successive pairs,

(11 + 13)/12     =  2
(17 + 19)/12     =  3
(29 + 31)/12     =  5
(41 + 43)/12     =  7

So there is a perfect matching here with the first four prime numbers. Though this pure pattern breaks down with the next few results, there is still a remarkably close relationship with the sequence of primes (with a difference of only 1 from the correct results in the sequence in evidence. So

(59 + 61)/12     = 10 (as opposed to 11)
(71 + 73)/12     = 12 (as opposed to 13)
(101 + 103)/12 = 17
(107 + 109)/12 = 18 (as opposed to 19)
(137 + 139)/12 = 23

Now with the next pair of twin primes the pattern breaks down somewhat with
(149 + 151)/12 = 25. So this value seems to be sticking out on its own (as compared to the next prime 29 . Then this is followed by,

(179 + 181)/12 = 30 (as opposed to 31) and
(191 + 193)/12 = 32
(197 + 199)/12 = 33

So we seem to have two additional results, here not matched by a corresponding prime before the next
(227 + 229)/12 = 38 (as opposed to 37) and
(239 + 241)/12 = 40 (as opposed to 41)

Then the next,

(269 + 271)/12 = 45 (which again appears as an additional results before the next,

(281 + 283)/12 = 47 and
(311 + 313)/12 = 52 (as opposed to 53).

Then we seem to bypass the primes i.e. 59, 61 and 67 before the nest two results

(419 + 421)/12 = 70 (as opposed to 71) and
(431 + 433)/12 = 72 (as opposed to 73)

And the next we have

(461 + 463) 12 = 77 (as opposed to 79)

So here the difference is 2, and the result is indicative of the fact that close relationship with the sequence of primes is slowly breaking down. So the remaining results up to 100 are

(521 + 523)/12 = 87
(569 + 571)/12 = 95

So again we seem to be missing a value for 83 while 87 and 95 differ from the next two primes 89 and 97 by 2 respectively.

However, it still seems remarkable that the results when dividing the sums of successive pairs of twin primes, should match the corresponding sequence of prime numbers so closely.