## Sunday, May 22, 2011

### Odd Numbered Integers (5)

We now look at the qualitative significance of the precise numerical values for the Riemann Zeta Function (with respect to negative odd integers of s).

As we earlier have seen with respect to positive even integer values of s, a definite pattern applies to the denominators of the rational fractions associated with these values.
In this context the denominator for an even integer will always be exactly divisible by the largest prime number up to s + 1 (and by no primes higher than this value).

However the reverse does not necessarily hold. In other words if the denominator is exactly divisible by s + 1, this does not entail that s + 1 is a prime number.
For example where s = 8, the denominator is 945. And 945 is in turn exactly divisible by s + 1 = 9. However 9 is not of course a prime number.

A similar - though more compelling sort of pattern - applies to the denominators of the rational values associated with the results of the Riemann Zeta Function (for negative odd integers).

If where s is negative the denominator of the result is always exactly divisible by s - 2 where the absolute value of s - 2 is prime.

Fro example the result of the Zeta Function where s = - 9 is - 1/132. Therefore the denominator here 132 is exactly divisible by - 11. And 11 is here prime.
However in this case, we can perhaps go further by suggesting that the converse is also true.

In other words if the denominator is in fact exactly divisible by s - 2, then this does seem to imply (on the basis of examination of all results of the Function for s up to - 200) that the absolute value of s - 2 is thereby prime.

In this way the Zeta Function for negative odd integral values can be seen as directly related to the generation of successive prime numbers.

Though it could not be suggested as a practical way to generate prime numbers, it is conjectured here that the entire set of prime numbers could be generated with reference to the division of the denominators of the Function (again for negative odd integral values of s) by s - 2. So once again wherever the denominator is exactly divisible by s - 2, then the absolute value of s - 2 is thereby prime.