RiemannSiegelTheta

The Riemann–Siegel theta function $\vartheta(t) = \operatorname{Im}\ln\Gamma(\tfrac14 + \tfrac{it}{2}) - \tfrac{t}{2}\ln\pi$, for real $t$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageRiemannSiegelTheta✓↗

RiemannSiegelTheta(t)the Riemann–Siegel theta function $\vartheta(t)$.

mpmathsiegeltheta(t)

Typed signatures

Type	Package	Applies
(number) -> number	analytic	always

Domain: Special functions

Details

• Reuses the existing log-gamma continuation (LogGamma) rather than deriving one — $\vartheta$ is just its imaginary part on the $\operatorname{Re} = \tfrac14$ line, minus the linear term.
• Real $t$ only; a complex argument stays symbolic. $\vartheta(0) = 0$ exactly (no N() needed) since $\ln\Gamma(\tfrac14)$ is real.
• Numeric otherwise: N(), or an inexact $t$, is required to reduce.

Plot

Examples

1 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{RiemannSiegelTheta}(1.5)$Out$-2.198\,190\,857\,379\,41$

live ↗

In$N(\mathrm{RiemannSiegelTheta}(10))$Out$-3.067\,074\,396\,289\,895\,4$

Properties #

live ↗

In$\mathrm{RiemannSiegelTheta}(0)$Out$0$

About

Examples	4
Checked against	Wolfram Language 4 agreempmath 4 agree
Crosswalk	Wolfram Languagempmath

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/riemann-siegel.ts

built directly on the LogGamma kernel; no new zeta or gamma evaluation of its own.

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

RiemannSiegelTheta[t]; mpmath.siegeltheta(t).

mappedmpmathpackages/oracle/src/mappings.ts

See also: RiemannSiegelZ, Zeta, LogGamma