## Friday, November 18, 2011

### Interesting Square Connection

If we just focus on the absolute value of the denominator of the Riemann's Zeta Function for s = - 1, - 3, - 5, - 7 and - 9, which are 12, 120, 252, 240 and 132 respectively we can find an interesting square connection.

So each of these numbers can be expressed as the product of two numbers which differ in ascending order by consecutive powers of 2.

So 12 = 4 * 3 with the difference (1) = 2^0.
120 = 12 * 10 with the difference (2) = 2^1.
252 = 18 * 14 with the difference (4) = 2^2.
240 = 20 * 12 with the difference (8) = 2^3.
132 = 22 * 6 with the difference (16) = 2^4.

After this the pattern begins to break down!

The absolute value corresponding to the denominator for s = - 11 = 32760. This can indeed be expressed as the product of two numbers that differ by a square of 2 but not (but not 2^5).

So 32760 = 182 * 180 with the difference (2) = 2^1.

However with the next number corresponding to s = - 13, no such relationship exists as between the product of two numbers i.e. involving the difference of a square of 2.