Riemann's (and the Hurwitz) Zeta Function, With Derivatives

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 $\geq 1$. 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 $$\sum_{n=1}^{\infty} 1 / n^s.$$ 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 $$\sum_{n=0}^{\infty} 1 / (A + n)^s.$$ 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.

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.

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.

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.

Author

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

Note

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

Examples

zeta(2:10)
#> [1] 1.644934 1.202057 1.082323 1.036928 1.017343 1.008349 1.004077 1.002008
#> [9] 1.000995

if (FALSE) { # \dontrun{
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") } # }

zeta(2) - pi^2 / 6     # Should be 0
#> [1] 0
zeta(4) - pi^4 / 90    # Should be 0
#> [1] 4.440892e-16
zeta(6) - pi^6 / 945   # Should be 0
#> [1] 0
zeta(8) - pi^8 / 9450  # Should be 0
#> [1] 4.440892e-16
zeta(0, deriv = 1) + 0.5 * log(2*pi)  # Should be 0
#> [1] 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
#> [1] -2.791795e-12
zeta(0.5, deriv = 1) + 3.92264613  # Should be 0
#> [1] -9.209153e-09
zeta(2.0, deriv = 1) + 0.93754825431  # Should be 0
#> [1] -5.84377e-12