Graphical Analysis

D. Baruth, x87@iging.com

The procedure _ZetaX^{[1]}evaluates Riemann's Zeta functionζ(x)for real arguments using different algorithms and internal tables^{[2]}, utilizing the x87 FPU's 80-bit registers.^{[3]}Evaluation time varies between 0.25 µs (for fetching a value from the table) and 95 µs (for calculating the function atx≈ 1). The relative error ε is kept below 10^{-18}.

ζ(|x| ≤ 16)

The characteristics for negative arguments is emphasized in the following graph:

ζ(x < 0)

Atx = 0the value ofζis exactly -½. Near zero the procedure converges relatively slow (83 µs) but the error can be kept small, nonetheless:

*ζ(x ≈ 0)*

- The proprietary procedure
_ZetaXcan be called, through appropriate interfaces, by any standard X86/87 compiler (e.g. FORTRAN, C, etc).- Handbook of Mathematical Functions, Abramowitz & Stegun, Dover Publications, fifth printing, p807.
- Calculations were performed on an AMD Athlon XP 1700+, 1.47 GHz, microprocessor. Computer programs were written in Assembly (MASM 5.1).

Copyright Dan Baruth © 2006.
All rights reserved.