RiemannSiegelZ

The Riemann–Siegel function $Z(t) = e^{i\vartheta(t)}\zeta(\tfrac12 + it)$ , real-valued for real $t$ by construction. Provided by @enumeratio/analytic.

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

RiemannSiegelZ(t)the Riemann–Siegel Z-function.

Typed signatures

Type	Package	Applies
`(number) -> number`	`analytic`	always

Domain: Special functions

Details

Reuses RiemannSiegelTheta and the generalized-zeta kernel (Zeta/HurwitzZeta at $a=1$ ); only the phase rotation onto the real line is added.
Numeric via N() or an inexact $t$ , same as RiemannSiegelTheta. Below $|{\rm Re}\,t| \approx 100$ (real $t$ : 2000) it runs the Hurwitz zeta and log-gamma kernels; past that, the Riemann–Siegel formula with Gabcke's remainder terms ( $C_0$ to $C_{30}$ ), continued to complex $t$ with $|{\rm Im}\,t| \le \tfrac12\sqrt{|{\rm Re}\,t|}$ and carried in BigDecimal for the phase, which a double loses with $t$ . That is a double's digits for any $|{\rm Re}\,t|$ up to $10^{10}$ , where the zeta sum's phases lose a relative $10^{-9}$ by $t = 10^6$ .
N(…, d) past a double's digits uses the same formula as far as its terms reach (its series is asymptotic): about 60 digits at $t = 2\cdot10^4$ , 70 at $10^5$ , 90 at $10^6$ . Below that it falls back to the bignum Hurwitz zeta and log-gamma kernels (real $t$ up to $10^5$ , complex up to $2\cdot10^4$ ), and past both it stays unevaluated rather than print fewer digits than were asked. N's refinement loop reads the value at 20 to 40 digits past $d$ , so $d$ is that much less than these.
The sign of $Z$ on the real line is what RiemannZetaZero's zero-finder scans for.

Examples

1 more kept as data, checked against the oracles

Basic #

In

\mathrm{RiemannSiegelZ}(1.5)

Out

-0.595\,568\,336\,782\,887

In

\mathrm{RiemannSiegelZ}(20.5)

Out

0.599\,328\,702\,514\,751\,3

In

\mathrm{FullSimplify}(\operatorname{Series}(\mathrm{RiemannSiegelZ}(x), x, 0, 2))

Out

\mathrm{RiemannSiegelZ}(0)+x\mathrm{RiemannSiegelZ}^{\prime}(0)+\frac{1}{2}(\mathrm{RiemannSiegelZ}^{\prime\prime}(0)x^2)+O\left(x^3\right)

Scope #

A real-valued function evaluated at $\frac13$

In

N(\mathrm{RiemannSiegelZ}(\frac{1}{3}))

Out

-1.232\,186\,779\,063\,262\,3

An even function, so the value at $-12$ matches $Z(12)$

In

N(\mathrm{RiemannSiegelZ}(-12))

Out

-1.259\,888\,834\,124\,469\,6

In

N(\mathrm{RiemannSiegelZ}(\frac{5}{4}), 50)

Out

-0.650\,818\,895\,379\,208\,837\,126\,984\,179\,609\,387\,325\,917\,093\,440\,350\,87

In

N(\mathrm{RiemannSiegelZ}(\frac{4}{3}), 20)

Out

-0.629\,639\,150\,443\,481\,539\,65

In

N(\mathrm{RiemannSiegelZ}(4+i))

Out

-0.589\,299\,974\,971\,081\,3-0.104\,133\,167\,496\,573\,24i

$Z(10)$ , real and negative

In

N(\mathrm{RiemannSiegelZ}(10))

Out

-1.549\,194\,546\,181\,024\,1

In

\operatorname{Series}(\mathrm{RiemannSiegelZ}(x), x, 0, 2)

Out

\mathrm{RiemannSiegelZ}(0)+x\mathrm{RiemannSiegelZ}^{\prime}(0)+\frac{1}{2}(\mathrm{RiemannSiegelZ}^{\prime\prime}(0)x^2)+O\left(x^3\right)

In

\mathrm{FullSimplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{RiemannSiegelZ}(x), x, \mathrm{x0}, 2)))

Out

\frac{1}{2}(\mathrm{RiemannSiegelZ}^{\prime\prime}(\mathrm{x0})(x-\mathrm{x0})^2)+(x-\mathrm{x0})\mathrm{RiemannSiegelZ}^{\prime}(\mathrm{x0})+\mathrm{RiemannSiegelZ}(\mathrm{x0})

Next to the real axis the imaginary part of $Z$ is $y\,Z'(t)$ : here $-10^{-50}$ times $Z'(10^5) \approx 16.23$ , with all its digits

In

N(\Im(\mathrm{RiemannSiegelZ}(100\,000+\frac{-i}{10^{50}})), 20)

Out

-1.623\,035\,867\,336\,958\,674\,4\cdot10^{-49}

Properties #

$Z(0) = \zeta(1/2)$

In

N(\mathrm{RiemannSiegelZ}(0))

Out

-1.460\,354\,508\,809\,586\,8

In

\mathrm{FullSimplify}(ζ(i t+\frac{1}{2})-\mathrm{RiemannSiegelZ}(t)\exp(-i\mathrm{RiemannSiegelTheta}(t)))

Out

ζ(i t+\frac{1}{2})-\mathrm{RiemannSiegelZ}(t)\exp(-i\mathrm{RiemannSiegelTheta}(t))

About

Examples	15
Checked against	Wolfram Language 13 agree2 differ
Crosswalk	Wolfram Language

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

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

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

RiemannSiegelZ[t]; mpmath.siegelz(t).

See also: RiemannSiegelTheta, Zeta, RiemannZetaZero