## Monday, May 30, 2016

### Approximating ζ(s) between – 1 and – 2

We need to extend approximations for ζ(s), before suggesting a simple way of then achieving an approximation for any negative value.

Taking the central value – 1.5 (between – 1 and – 2),

ζ(– 1.5)  = – .025485...

This is well approximated by e – π(k) * 2k =  – .025408... (where k = 1.5)

I then attempted to obtain approximations based on this formula replacing the power (or exponent) with respect to both e and 2 with the corresponding value of k.

This works well down to k = 1.

However it is much more problematic in the other direction.

Here rather than changing k successively upwards from 1.6 to 2, it works better to decrease in again reaching k = 1 for s = – 2.

However even here the approximation proves very inaccurate as we approach s =  – 2, where ζ(– 2) = 0.

Once again I give a table showing values for s, ζ(s), approximation for ζ(s) and % relative accuracy.

 s ζ(s) Approximation % relative accuracy – 1.1 – .067981 – .067657 99.5 – 1.2 – .054788 – .052964 96.7 – 1.3 – .043464 – .041461 95.4 – 1.4 – .033764 – .032457 96.1 – 1.5 – .025485 – .025408 99.7 – 1.6 – .018448 – .017315 93.9 – 1.7 – .012505 – .011800 94.3 – 1.8 – .007522 – .008041 93.5 – 1.9 – .003387 – .005480 61.8 – 2.0 0 – .003734 0

With the exception of the final two values, the approximations are quite accurate. It seems that the negative even values for s, where ζ(s) = 0, operate as attractors in their vicinity, upsetting previous trends.