# zeta: Riemann Zeta Function In pracma: Practical Numerical Math Functions

## Description

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

## Usage

 `1` ```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.

`gammaz`, `eta`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```## 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) ```

### Example output

```
```

pracma documentation built on June 21, 2017, 9:01 a.m.