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AdvancedComplexMathRiemannZeta Method

Computes the Riemann zeta function for complex values.

Namespace:  Meta.Numerics.Functions
Assembly:  Meta.Numerics (in Meta.Numerics.dll) Version: 4.1.4
public static Complex RiemannZeta(
	Complex z


Type: Meta.NumericsComplex
The argument.

Return Value

Type: Complex
The value of ζ(z).

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.

See Also