zeta {VGAM} | R Documentation |
Computes Riemann's zeta function and its first two derivatives.
zeta(x, deriv = 0)
x |
A complex-valued vector/matrix whose real values must be >= 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.
|
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) * 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.
A vector/matrix of computed values.
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.
Estimation of the parameter of the zeta distribution can be achieved
with zetaff
.
T. W. Yee, with the help of G. J. Tee.
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.
zeta(2:10) ## Not run: x = seq(1.2, 10, len=100) plot(x, zeta(x), type="l", las=1, xlim=c(-12,10), ylim=c(-1,4), col="red") x = seq(-12, 0.8, len=100) lines(x, zeta(x), col="red") abline(v=0, h=c(0,1), lty="dashed") # Close up plot x = seq(-14, -0.4, len=100) plot(x, zeta(x), type="l", las=1, col="red") abline(v=0, h=0, lty="dashed") # Plot of the first derivatives x = seq(0.04, 0.8, len=100) plot(x, zeta(x, deriv=1), type="l", las=1, col="blue", xlim=c(0.04,3), ylim=c(-6,0)) x = seq(1.2, 3, len=100) lines(x, zeta(x, deriv=1), col="blue") abline(v=0, h=0, lty="dashed") ## End(Not run) 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