Functions
float	dirichlet_eta (float s)

float	dirichlet_beta (float s)

float	dirichlet_lambda (float s)

float	singularity_in_id (NDArray[np.float64] y, float s, int dim)

float	epstein_zeta_00_mhalfmhalf_id (float nu)

float	epstein_zeta_m1m1_halfhalf_id (float nu)

float	epstein_zeta_m1m1_half0_id (float nu)

float	epstein_zeta_onehalf0sqrt3half_00_00 (float nu)

float	epstein_zeta_diag2sqrt242_0m1m1_4sqrt2th00 (float nu)

float	epstein_zeta_half000_0000_id (float nu)

float	min_errors_abs_error_rel (complex true, complex approx)

Variables
	dps

Detailed Description

Reference functions for benchmarking purposes.

Function Documentation

◆ dirichlet_beta()

float benchmark_functions.dirichlet_beta ( float s )

Compute the Dirichlet eta function for a given s.
Representation in Terms of the Hurwitz zeta function.

◆ dirichlet_eta()

float benchmark_functions.dirichlet_eta ( float s )

Compute the Dirichlet eta function for a given s.
Representation in terms of the Riemann Zeta function.

◆ dirichlet_lambda()

float benchmark_functions.dirichlet_lambda ( float s )

Compute the Dirichlet lambda function for a given s.
Representation in Terms of the Riemann zeta function.

◆ epstein_zeta_00_mhalfmhalf_id()

float benchmark_functions.epstein_zeta_00_mhalfmhalf_id ( float nu )

Compute the Epstein Zeta function for a given nu ≠ 2 and

x = [0, 0]
y = [- 1 / 2, - 1 / 2]
a = [[1, 0], [0, 1]]

Representation in terms of the Dirichlet Eta function
and the Dirichlet Beta function.

◆ epstein_zeta_diag2sqrt242_0m1m1_4sqrt2th00()

float benchmark_functions.epstein_zeta_diag2sqrt242_0m1m1_4sqrt2th00 ( float nu )

Compute the Epstein Zeta function for a given nu and

x = [0, -1, -1]
y = [-4*sqrt(2), 0, 0]
a = [[2sqrt(2), 0, 0], [0, 4, 0], [0, 0, 2]]

Representation in terms of the Dirichlet Beta function.

◆ epstein_zeta_half000_0000_id()

float benchmark_functions.epstein_zeta_half000_0000_id ( float nu )

Compute the Epstein Zeta function for a given nu ≠ 2 and

x = [1 / 2, 0, 0, 0]
y = [0, 0, 0, 0]
a = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]

Representation in terms of the Dirichlet Beta function
and the Zeta function.

◆ epstein_zeta_m1m1_half0_id()

float benchmark_functions.epstein_zeta_m1m1_half0_id ( float nu )

Compute the Epstein Zeta function for a given nu ≠ 2 and

x = [-1, -1]
y = [1 / 2, 0]
a = [[1, 0], [0, 1]]

Representation in terms of the Dirichlet Beta function
and the Zeta function.

◆ epstein_zeta_m1m1_halfhalf_id()

float benchmark_functions.epstein_zeta_m1m1_halfhalf_id ( float nu )

Compute the Epstein Zeta function for a given nu ≠ 2 and

x = [-1, -1]
y = [1 / 2, 1 / 2]
a = [[1, 0], [0, 1]]

Representation in terms of the Dirichlet Beta function
and the Zeta function.

◆ epstein_zeta_onehalf0sqrt3half_00_00()

float benchmark_functions.epstein_zeta_onehalf0sqrt3half_00_00 ( float nu )

Compute the Epstein Zeta function for a given nu and

x = [0, 0]
y = [0, 0]
a = [[1, 1/2], [0, sqrt(3)/2]]

Representation in terms of the (hurwitz) Zeta function.

◆ min_errors_abs_error_rel()

float benchmark_functions.min_errors_abs_error_rel	(	complex	true,
		complex	approx )

Return the minimum of absolute and relative error.

◆ singularity_in_id()

float benchmark_functions.singularity_in_id	(	NDArray[np.float64]	y,
		float	s,
		int	dim )

Compute the Singularity of the Epstein Zeta functions
when y goes to zero for any vector x, identity matrix a
and nu ≠ 2 + n for any natural number n including zero.

epstein_zeta_reg(x, y, id, nu) =
    exp(2 * Pi * i * x . y)
    * epstein_zeta(x, y, id, nu)
    - singularity(y, nu)

Functions

Variables