mathxlab.nt.zeta

Riemann zeta function and related numerical helpers.

Stability

Status: Experimental.

This project treats the documented names as the public surface, but details may still evolve. If you need strict API guarantees, add __all__ = [...] to each module and version releases accordingly.

Design notes

• Functions are designed for experiment-scale inputs (not cryptographic workloads).

• Prefer explicit parameters (e.g., n_max) for reproducibility.

Examples

Approximate ζ(2) via a partial series

from mathxlab.nt.zeta import zeta_series_partial
print(zeta_series_partial(2, 10_000))

Public API

Name	Kind	Summary
ZetaEvalSettings	class	Settings for zeta-related numerical evaluations.
mp_workdps	function	Temporarily set the mpmath precision (decimal digits).
zeta_series_partial	function	Compute the partial Dirichlet series for the Riemann zeta function.
eta_series_partial	function	Compute the partial Dirichlet eta series.
zeta_via_eta	function	Recover zeta(s) from eta(s) via the identity.
euler_product_partial	function	Compute a partial Euler product approximation of zeta(s).
chi_factor	function	Compute the factor chi(s) in the functional equation of zeta.
hardy_Z	function	Compute Hardy’s Z-function at height t.
riemann_von_mangoldt_count	function	Return the Riemann–von Mangoldt main term for N(T).

Reference

Classes

class mathxlab.nt.zeta.ZetaEvalSettings(dps=50)

Bases: object

Settings for zeta-related numerical evaluations.

Parameters:

dps – Decimal digits of precision used by mpmath during the computation.

Examples

>>> from mathxlab.nt.zeta import ZetaEvalSettings
>>> ZetaEvalSettings

Functions

mathxlab.nt.zeta.mp_workdps(dps)

Temporarily set the mpmath precision (decimal digits).

Parameters:

dps – Decimal digits of precision.

Yields:

None. The previous precision is restored on exit.

Examples

>>> from mathxlab.nt.zeta import mp_workdps
>>> mp_workdps

mathxlab.nt.zeta.zeta_series_partial(s, n_max, *, settings=None)

Compute the partial Dirichlet series for the Riemann zeta function.

This computes:

sum_{n=1..n_max} n^{-s}

Parameters:

• s – Complex exponent.

• n_max – Number of terms.

• settings – Optional evaluation settings.

Returns:

Complex partial sum as a Python complex number.

Examples

>>> from mathxlab.nt.zeta import zeta_series_partial
>>> round(zeta_series_partial(2.0, 2000).real, 3)
1.644

mathxlab.nt.zeta.eta_series_partial(s, n_max, *, settings=None)

Compute the partial Dirichlet eta series.

This computes:

eta(s) = sum_{n=1..n_max} (-1)^{n-1} n^{-s}

Parameters:

• s – Complex exponent.

• n_max – Number of terms.

• settings – Optional evaluation settings.

Returns:

Complex partial sum as a Python complex number.

Examples

>>> from mathxlab.nt.zeta import eta_series_partial
>>> eta_series_partial

mathxlab.nt.zeta.zeta_via_eta(s, eta_value)

Recover zeta(s) from eta(s) via the identity.

For s != 1:

zeta(s) = eta(s) / (1 - 2^{1-s})

Parameters:

• s – Complex argument.

• eta_value – Value of eta(s) (possibly a partial sum).

Returns:

Complex approximation of zeta(s).

Examples

>>> from mathxlab.nt.zeta import zeta_via_eta
>>> zeta_via_eta

mathxlab.nt.zeta.euler_product_partial(s, primes, *, settings=None)

Compute a partial Euler product approximation of zeta(s).

This computes:

prod_{p in primes} (1 - p^{-s})^{-1}

Parameters:

• s – Complex argument.

• primes – Iterable of prime numbers.

• settings – Optional evaluation settings.

Returns:

Complex approximation as a Python complex.

Examples

>>> from mathxlab.nt.zeta import euler_product_partial
>>> euler_product_partial

mathxlab.nt.zeta.chi_factor(s, *, settings=None)

Compute the factor chi(s) in the functional equation of zeta.

One convenient form is:

zeta(s) = chi(s) * zeta(1 - s)

where:

chi(s) = 2^s * pi^{s-1} * sin(pi s / 2) * Gamma(1 - s)

Parameters:

• s – Complex argument.

• settings – Optional evaluation settings.

Returns:

chi(s) as a Python complex.

Examples

>>> from mathxlab.nt.zeta import chi_factor
>>> chi_factor

mathxlab.nt.zeta.hardy_Z(t, *, settings=None)

Compute Hardy’s Z-function at height t.

mathxlab.nt.zeta.riemann_von_mangoldt_count(T)

Return the Riemann–von Mangoldt main term for N(T).