Riemann's Zeta function  ζ(x)
Graphical Analysis



D. Baruth, x87@iging.com




The procedure  _ZetaX[1]  evaluates Riemann's Zeta function  ζ(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 at  x ≈ 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)


At  x = 0  the value of  ζ  is exactly .  Near zero the procedure converges relatively slow (83 µs) but the error can be kept small, nonetheless:

ζ(x ≈ 0)



  1. The proprietary procedure  _ZetaX  can be called, through appropriate interfaces, by any standard X86/87 compiler (e.g. FORTRAN, C, etc).
  2. Handbook of Mathematical Functions, Abramowitz & Stegun, Dover Publications, fifth printing, p807.
  3. 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.