VGAM (version 0.9-2)

zeta: Riemann's Zeta Function

Description

Computes Riemann's zeta function and its first two derivatives.

Usage

zeta(x, deriv = 0)

Arguments

x

A complex-valued vector/matrix whose real values must be $\geq 1$. Otherwise, if x may be real. 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.

Value

• A vector/matrix of computed values.

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< code=""> because currently the gamma function does not handle complex arguments.

Details

While the usual definition involves an infinite series, 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 formula used here for analytic continuation is $$\zeta (s)=2^s{\pi }^{s-1}\sin (\pi s/2)\mathrm{\Gamma }(1-s)\zeta (1-s).$$ This is actually one of several formulas, but this one was discovered by Riemann himself and is called the functional equation.

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, lerch, gamma.

Examples

zeta(2:10)

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")

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

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 zero
zeta(4) - pi^4 / 90    # Should be zero
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