Description Usage Arguments Details Value References See Also Examples

Numerical computation of the Epstein zeta function.

1 |

`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 |

`prepare` |
a logical or a list. |

`norm` |
logical. Should the phase be normalized? Default: TRUE. See details below. |

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.

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.

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

VecLat-class

1 2 |

```
Loading required package: gsl
Loading required package: R2Cuba
[1] 2.110572
[1] 8.892745
```

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