# ezeta: Epstein zeta function In pgs: Precision of Geometric Sampling

## Description

Numerical computation of the Epstein zeta function.

## Usage

 `1` ```Ezeta(s, vlat, h = rep(0, vlat@dimspace), L = 3, prepare = FALSE, norm = TRUE) ```

## Arguments

 `s` the exponent parameter as a numeric. See details below. `vlat` a vector lattice as a VecLat-class object. `h` a phase vector or a matrix of phase column vectors. `L` the stopping criterion for the numerical approximation of the Epstein zeta function. Default: 3. Increase `L` for better precision. `prepare` a logical or a list. `norm` logical. Should the phase be normalized? Default: TRUE. See details below.

## Details

The Epstein zeta function is a multidimensional version of the Riemann zeta function defined as the sum of

exp(-2*pi*I*<h,x>)/|x|^s

for all non-null vectors x of the lattice.

When considered as a function of the phase `h`, the Epstein zeta function is invariant under any translation by a lattice vector. The phase vector `h` provided to `Ezeta` must lie in the fundamental tile of the vector lattice `vlat`. If `norm` is TRUE, `h` is automatically normalized.

The algorithm used for computation of the Epstein zeta function is provided in a paper by Richard E. Crandall, see reference below. In this implementation, all preliminary computations not depending on the phase `h` can be made separately.

## Value

If `prepare` is FALSE, the result as a numeric. If `prepare` is TRUE, preliminary computations not depending on the phase are returned as a list. If `prepare` is a list as computed when `prepare` is TRUE, the final result as a numeric.

## References

Crandall, R.E. (1998). Fast evaluation of Epstein zeta functions. Manuscript. http://www.reed.edu/~crandall/papers/epstein.pdf

VecLat-class

## Examples

 ```1 2``` ```Ezeta(3,RectLat2(),h=c(1.1,3.8)) Ezeta(3,HexLat2()) ```

### Example output

```Loading required package: gsl