R: Riemann's Zeta Function

zeta {VGAM} | R Documentation

## Riemann's Zeta Function

### Description

Computes Riemann's zeta function and its first two derivatives. Also can compute the Hurwitz zeta function.

### Usage

    zeta(x, deriv = 0, shift = 1)

### Arguments

`x` | A complex-valued vector/matrix whose real values must be >= 1. Otherwise, `x` may be real. It is called s below. If `deriv` is 1 or 2 then `x` must be real and positive.
`deriv` | An integer equalling 0 or 1 or 2, which is the order of the derivative. The default means it is computed ordinarily.
`shift` | Positive and numeric, called A below. Allows for the Hurwitz zeta to be returned. The default corresponds to the Riemann formula.

### Details

The (Riemann) formula for real s is

sum_{n=1}^Inf 1 / n^s.

While the usual definition involves an infinite series that converges when the real part of the argument is > 1, more efficient methods have been devised to compute the value. In particular, this function uses Euler-Maclaurin summation. Theoretically, the zeta function can be computed over the whole complex plane because of analytic continuation.

The (Riemann) formula used here for analytic continuation is

zeta(s) = 2^s * pi^(s-1) * sin(pi*s/2) * gamma(1-s) * zeta(1-s).

This is actually one of several formulas, but this one was discovered by Riemann himself and is called the functional equation.

The Hurwitz zeta function for real s > 0 is

sum_{n=0}^Inf 1 / (A + n)^s.

where 0 < A is known here as the `shift`. Since A=1 by default, this function will therefore return Riemann's zeta function by default. Currently derivatives are unavailable.

### Value

The default is a vector/matrix of computed values of Riemann's zeta function. If `shift` contains values not equal to 1, then this is Hurwitz's zeta function.

### Warning

This function has not been fully tested, especially the derivatives. In particular, analytic continuation does not work here for complex `x` with `Re(x)<1` because currently the `gamma` function does not handle complex arguments.

### Note

Estimation of the parameter of the zeta distribution can be achieved with `zetaff`.

### Author(s)

T. W. Yee, with the help of Garry J. Tee.

### References

Riemann, B. (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse. Monatsberichte der Berliner Akademie, November 1859.

Edwards, H. M. (1974). Riemann's Zeta Function. Academic Press: New York.

Markman, B. (1965). The Riemann zeta function. BIT, 5, 138–141.

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications Inc.

### See Also

`zetaff`, `oazeta`, `oizeta`, `otzeta`, `lerch`, `gamma`.

### Examples

    zeta(2:10)

    ## Not run:
    curve(zeta, -13, 0.8, xlim = c(-12, 10), ylim = c(-1, 4), col = "orange",
          las = 1, main = expression({zeta}(x)))
    curve(zeta, 1.2,  12, add = TRUE, col = "orange")
    abline(v = 0, h = c(0, 1), lty = "dashed", col = "gray")

    curve(zeta, -14, -0.4, col = "orange", main = expression({zeta}(x)))
    abline(v = 0, h = 0, lty = "dashed", col = "gray")  # Close up plot

    x <- seq(0.04, 0.8, len = 100)  # Plot of the first derivative
    plot(x, zeta(x, deriv = 1), type = "l", las = 1, col = "blue",
         xlim = c(0.04, 3), ylim = c(-6, 0), main = "zeta'(x)")
    x <- seq(1.2, 3, len = 100)
    lines(x, zeta(x, deriv = 1), col = "blue")
    abline(v = 0, h = 0, lty = "dashed", col = "gray")
    ## End(Not run)

    zeta(2) - pi^2 / 6     # Should be 0
    zeta(4) - pi^4 / 90    # Should be 0
    zeta(6) - pi^6 / 945   # Should be 0
    zeta(8) - pi^8 / 9450  # Should be 0
    zeta(0, deriv = 1) + 0.5 * log(2*pi)  # Should be 0
    gamma0 <-  0.5772156649
    gamma1 <- -0.07281584548
    zeta(0, deriv = 2) -
      gamma1 + 0.5 * (log(2*pi))^2 + pi^2/24 - gamma0^2 / 2  # Should be 0
    zeta(0.5, deriv = 1) + 3.92264613  # Should be 0
    zeta(2.0, deriv = 1) + 0.93754825431  # Should be 0

[Package VGAM version 1.1-3 Index]