Wednesday, February 1, 2012

Approximation Formulae for Zeta Function (Negative Odd Integers)

These are some approximation formulae that I devised for my own use.

Let s = negative odd integer

Now to simplify the following expression a little I write k = - (s + 1)/2

So for example when s = -3, k = - (- 3 + 1)/2 = 1.

ζ(s-2)/ζ(s) → .1 + 11k/{[2(pi^2)]} + k(k–3)/(pi^2)

Thus if we already know the value for ζ(s), we can use it to approximate the corresponding value for ζ(s-2).

We can also give an approximating formula for difference of ratios

ζ(s-4)/ζ(s-2) - ζ(s-2)/ζ(s) → 11/{2[(pi^2)]} + 2(k-1)/(pi^2)

Also we can provide an even simpler formula for approximating the difference of difference as between successive ratios!

{ζ(s-6)/ζ(s-4) – ζ(s-4)/ζ(s-2)} – {ζ(s-4)/ζ(s-2) – ζ(s-2)/ζ(s)} → 2/(pi^2).

Here is one other formula that involves a link with even values.

So if we take s now as an even integer, then

│{ζ(1-s)}^(1/s)/{ζ1-s-k)^[1/(s-k)]}│ → s/(s-k)

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