# zetaUC: The Zeta Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

Density, distribution function, quantile function and random generation for the zeta distribution.

## Usage

 ```1 2 3 4``` ```dzeta(x, shape, log = FALSE) pzeta(q, shape, lower.tail = TRUE) qzeta(p, shape) rzeta(n, shape) ```

## Arguments

 `x, q, p, n` Same as `Poisson`. `shape` The positive shape parameter p. `lower.tail, log` Same meaning as in `Normal`.

## Details

The density function of the zeta distribution is given by

y^(-s-1) / zeta(s+1)

where s>0, y=1,2,…, and zeta is Riemann's zeta function.

## Value

`dzeta` gives the density, `pzeta` gives the distribution function, `qzeta` gives the quantile function, and `rzeta` generates random deviates.

## Note

`qzeta()` runs slower and slower as `shape` approaches 0 and `p` approaches 1. The VGAM family function `zetaff` estimates the shape parameter s.

T. W. Yee

## References

Johnson N. L., Kotz S., and Balakrishnan N. (1993). Univariate Discrete Distributions, 2nd ed. New York: Wiley.

`zeta`, `zetaff`, `Oazeta`, `Oizeta`, `Otzeta`.
 ``` 1 2 3 4 5 6 7 8 9 10``` ```dzeta(1:20, shape = 2) myshape <- 0.5 max(abs(pzeta(1:200, myshape) - cumsum(1/(1:200)^(1+myshape)) / zeta(myshape+1))) # Should be 0 ## Not run: plot(1:6, dzeta(1:6, 2), type = "h", las = 1, col = "orange", ylab = "Probability", main = "zeta probability function; orange: shape = 2; blue: shape = 1") points(0.10 + 1:6, dzeta(1:6, 1), type = "h", col = "blue") ## End(Not run) ```