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)} * 2^k =  – .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.