Unexpectedly this morning, while trying out an insight that struck me yesterday, I seemed to have detected a very interesting pattern that governs the denominators of values of the Riemann Zeta Function for negative odd integer values.
This pattern relates to divisibility of the denominator by the first two perfect numbers 6 and 28 and can be stated succinctly as follows.
(i) The denominator of such values is always divisible by 6.
(ii) in every 3rd case the denominator is divisible by both 6 and 28 (and only in such a case).
(iii) The denominator does not appear to be divisible by any other perfect numbers.
For example the 1st zeta result where s = - 1 is - 1/12 and the denominator is clearly divisible by 6.
The 2nd zeta result where s = - 3 is 1/120 and again the denominator is divisible by 6.
The 3rd zeta result where s = - 5 is - 1/252 and the denominator here is divisible by both 6 and 28.
And this trend continues. So the denominators (240 and 132) for both s = - 7 and s = - 9 are divisible by 6, whereas the denominator for s = - 11 (32760) which is the 3rd in the sequence, is divisible by both 6 and 28. Indeed in this case it is divisble by 6 * 28. However that is not generally the case!
Using zeta results compiled in Mc Gill University this trend can be verified for the first relevant 100 zeta values (i.e. up to s = -199).
It should be also stated that this numerical behavioural characteristic does not extend to denominators of the zeta function for positive even integer values of s!
However with respect to any qualitative interpretation of the meaning of such results it is perhaps too early to speculate.
Indeed even more dramatic numerical patterns exist with respect to the denominator of these zeta values (for negative odd integers).
As we have seen each successive value is divisible by 6 (i.e. 2 * 3).
Then every 2nd successive denominator value is divisible by 5; every 3rd succesive denominator nvalue is divisible by 7; every 5th successive denominator value is divisible by by 11; every 6th successive denominator value is divisible by 13; every 8th successive denominator value is divisible by 17; every 9th successive denominator value by 19; every 11th successive denominator value is divisible by 23; every 14th successive denominator value is divisible by 29 and so on.
In other words where the absolute value of s is prime, every {|s - 1|/2)th denominator value in the zeta sequence for negative odd integral values is thereby divisible by |s|.
Put another way if therefore every {|s - 1|/2)th denominator value is divisible by |s|, then |s| is prime.
For example if |- 31| is prime then every 15th value in the zeta sequence should be divisible by 31
Now the absolute value of the denominator is the first of these cases (15th value in sequence) is 85932 which is divisible 31.
Now for every further 15th value in sequence the absolute value of denominator will be divisible by 31.
For example the absolute value for the denominator of 30th value is 3407203800 which once again is divisible by 31.
Therefore we could conclude from this that 31 is a prime number!
Not alone does this pattern appear to hold unbiversally but equally for all absolute prime values of s > 3, the only time when the demominator is divisible by |s| is for the {|s - 1|/2)th denominator value in the sequence.
Therefore we could safely conclude for this value of |s| where = 31, that 31 is indeed a prime number.
And of course this would hold for all other values of |s| where the same principle applies!
Put finally yet another way if s = 2, 4 ,6, 8,....
then for zeta (1 - s), where the value of (s + 1) is prime, the absolute value of denominator will be divisible by all factors of s where with the addition of 1 are prime (and only these factors).
So one again for example when s = 36, zeta (1 - s) is zeta (- 35).
In this case s + 1 = 37 which is prime. Now all the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Now with addition of 1 in each case we get 2, 3, 4, 5, 7, 10, 13, 19 and 37.
However since 4 and 10 are not prime we can exclude these numbers.
Theerfore the denominator of zeta (-35) i.e. 69090840 is divisible by 2, 3, 5, 7, 13, 19 and 37 (and only these prime numbers).
And what is remarkable is that when the denominator is divided by the product of all these prime factors the result is s.
Therfore 69090840/(2 * 3 * 5 * 7 * 13 * 19 * 37) = 36.
Though this final result does not universally hold it does so in some cases i.e. when the denominator is divided by product of all prime factors the result is s.
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