pracma (version 1.7.0)

zeta: Riemann Zeta Function

Description

Riemann's zeta function valid in the entire complex plane.

Usage

zeta(z)

Arguments

z

Real or complex number or a numeric or complex vector.

Value

• Returns a complex vector of function values.

Details

Computes the zeta function for complex arguments using a series expansion for Dirichlet's eta function.

Accuracy is about 13 significant digits for abs(z)<100< code="">, drops off with higher absolute values.

References

Zhang, Sh., and J. Jin (1996). Computation of Special Functions. Wiley-Interscience, New York.

See Also

gammaz, eta

Examples

##  First zero on the critical line s = 0.5 + i t
x <- seq(0, 20, len=1001)
z <- 0.5 + x*1i
fr <- Re(zeta(z))
fi <- Im(zeta(z))
fa <- abs(zeta(z))
plot(x, fa, type="n", xlim = c(0, 20), ylim = c(-1.5, 2.5),
     xlab = "Imaginary part (on critical line)", ylab = "Function value",
     main = "Riemann's Zeta Function along the critical line")
lines(x, fr, col="blue")
lines(x, fi, col="darkgreen")
lines(x, fa, col = "red", lwd = 2)
points(14.1347, 0, col = "darkred")
legend(0, 2.4, c("real part", "imaginary part", "absolute value"),
       lty = 1, lwd = c(1, 1, 2), col = c("blue", "darkgreen", "red"))
grid()