pgs (version 0.4-0)

Ezeta: Epstein zeta function

Description

Numerical computation of the Epstein zeta function.

Usage

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.

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.

Details

The Epstein zeta function is a multidimensional version of the Riemann zeta function defined as the sum of $$\frac{\exp(-2\pi I \langle h,x \rangle )}{\|x\|^s}$$ 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.

References

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

See Also

VecLat-class

Examples

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