More on the Significance of ζ– 1) = – 1/12!

In my companion blog "Spectrum of Mathematics", I have shown how a fascinating formula of Ramanujan for 1/24 can be used to "convert" the existing series - in an intuitively accessible linear manner - for ζ(s), where s is a negative odd integer.

The standard "conversion" formula is given for the case of ζ(– 13).

Now in linear terms, 

ζ(– 13) = 1¹³ + 2¹³ + 3¹³ + ...., which diverges to infinity. However we know that the correct answer (in holistic terms) = – 1/12.

So the "converted" series, i.e.

1¹³/{(1 –  e^2π )/2} + 2¹³/{(1 –  e^4π )/2} + 3¹³/{(1 –  e^6π )/2} +...    = – 1/12,  now gives the correct answer from the standard linear perspective.

The remarkable fact is that this formula very much works in cyclical fashion in the manner of a 12 hr. clock for the denominator (with the hours referring to the common dimensional power to which each of the natural numbers in the series is raised).

Thus when for example we go back "12 hours" we find  that,

ζ(– 1) = 1¹ + 2¹ + 3¹ + ....,

And the correct "converted" series for ζ(– 1) is given by the same "converted" series for ζ(– 13), i.e.

1¹³/{(1 –  e^2π )/2} + 2¹³/{(1 –  e^4π )/2} + 3¹³/{(1 –  e^6π )/2} +...    = – 1/12.

When we now move forward "12 hours" we find that

ζ(– 25) = 1²⁵ + 2²⁵ + 3²⁵ + ....,  which  =  – 657931/12 (from the correct holistic perspective).

So the linear "conversion" for this series is given as,

657931[1¹³/{(1 –  e^2π )/2} + 2¹³/{(1 –  e^4π )/2} + 3¹³/{(1 –  e^6π )/2}] +...
= – 1/12.

Now the "12 hour" clock does not always work in this manner for the denominator. It does work again where s = – 37, but then breaks down for  s = – 49, with the denominator for ζ(– 49) = – 132.

However for all  "12 hour" cycles then up to (and including) s = – 97, the denominator of ζ(s) = – 12.

The importance of 12 is also in evidence with respect to the fact that it seems that in all cases (where s is a negative odd integer), that the denominator of ζ(s) is divisible by 12. I have also been aware for some time here of an apparent connection with twin primes. So apart from the first twin primes i.e. 3 and 5, the sum of all other twin prime pairings appears to be divisible by 12!

However it is another remarkable feature - with respect to the denominators of ζ(s) - where again s is  negative odd integer, that I wish to concentrate on here!

In conventional terms, we can determine that a number is prime if it has no factors (other than itself and 1). However, as is known so well, it becomes progressively difficult to test for primes in this manner (where the number is very large).

In fact, safe encryption systems - on which e-commerce so much depends - are based on the inherent difficulty of factorising very large numbers!

However the "Alice in Wonderland" world of the Riemann zeta function, where conventional expectations with respect to results are turned inside-out for negative values of s < 0, in principle offers a complementary opposite means for testing for number primality.

Once again in conventional linear terms, we demonstrate that a number is prime by showing that it has no other factors (other than the number itself and 1).

Then in complementary circular terms, we can demonstrate that a number is prime by showing in a certain unique manner - enabled by the Riemann zeta function - that it is a factor of a composite number!

In fact the "golden rule" for establishing such primality can be stated quite simply.

In Riemann's functional equation, a relationship is established as between values of ζ(s) and ζ(1 – s).

So when s is a positive even, 1 – s is thereby a negative odd integer.

Now the "golden rule" is as follows.

If the denominator of ζ(1 – s) is divisible by 1 + s, then 1 + s  is a prime number; also if the denominator of ζ(1 – s) is not divisible by (1 + s), then 1 + s  is not a prime number. 

When s = 2 the denominator of ζ(– 1) i.e. 12 is divisible by 3. Therefore 3 is prime.

When s = 4, the denominator of ζ(– 3) i.e. 120 is divisible by 5. Therefore 5 is prime.

When s = 6, the denominator of ζ(– 5) i.e. 252 is divisible by 7. Therefore 7 is prime.

Then when s = 8, the denominator of ζ(– 7) i.e. 240 is not divisible by 9. Therefore 9 is not prime!

When s = 10, the denominator of ζ(– 9) i.e. 132 is divisible by 11. Therefore 11 is prime.

When s = 12 the denominator of ζ(–11) i.e. 32760 is divisible by 13. Therefore 13 is prime.

Then when s = 14, the denominator of ζ(–11) i.e. 12 is not divisible by 15. Therefore 15 is not prime!

And my contention is that we can continue on indefinitely in this manner. I have found no exceptions in this procedure to s = 200, which is as far as the available tables enable me to test.

In fact the general rule also holds for the one case where a prime can be even (= 2).
For when s = 1, the denominator of ζ(0) i.e. 2 is divisible by 2!


Now of course, this does not provide a practical way of testing for large prime numbers as the calculation of the corresponding denominators of ζ(1 – s) becomes prohibitively cumbersome.

However what remains fascinating is that in principle it does provide a means of testing for primes which completely inverts the customary procedure. And this in turn is valuable to point out, as it demonstrates from yet another perspective the truly complementary nature of both ζ(s) and ζ(1 – s) which require analytic and holistic interpretation with respect to each other.

And this can only be properly understood in a dynamic interactive context, where both analytic and holistic aspects - that are relatively quantitative and qualitative with respect to each other - are clearly recognised.

In addition, it can be stated that when ζ(1 – s) is divisible by (1 + s), this then represents the largest prime by which ζ(1 – s) is divisible.

One can readily make comparisons with the denominator of ζ(s) where again s even. Here the denominator of ζ(s) entails the product of all primes (where repetition of the same prime is allowed) from 2 to 1 + s (where s is a power of 2) and from 3 to 1 + s in all other cases.
However we cannot use the denominator here to universally establish whether a number is prime!