R: Riemann Zeta Function

zeta {pracma} | R Documentation

## 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.

### 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`, drops off with higher absolute values.

### Value

Returns a complex vector of function values.

### Note

Copyright (c) 2001 Paul Godfrey for a Matlab version available on Mathwork's Matlab Central under BSD license.

### 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
    ## Not run:
    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()
    ## End(Not run)

[Package pracma version 2.0.4 Index]