AdvancedComplexMathRiemannZeta Method

public static Complex RiemannZeta(
	Complex z
)

Public Shared Function RiemannZeta ( 
	z As Complex
) As Complex

public:
static Complex RiemannZeta(
	Complex z
)

static member RiemannZeta : 
        z : Complex -> Complex 

Parameters

z

Type: Meta.NumericsComplex
The argument.

Return Value

As the imaginary part of the argument increases, the computation of the zeta function becomes slower and more difficult. The computation time is approximately proportional to the imaginary part of z. The result also slowly looses accuracy for arguments with very large imaginary parts; for arguments with z.Im of order 10^D, approximately the last D digits of the result are suspect.

The image below shows the complex ζ function near the origin using domain coloring. You can see the first non-trivial zeros at (1/2, ±14.13...) as well as the trivial zeros along the negative real axis.

Reference

Other Resources