MultiZetaValue

The depth-2 multiple zeta value $\zeta(s_1, s_2) = \sum_{n_1 > n_2 \ge 1} n_1^{-s_1} n_2^{-s_2}$, for integers $s_1, s_2 \ge 2$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimMultiZetaValue✓↗Fungrim entry

MultiZetaValue(s1, s2)the depth-2 Euler sum $\zeta(s_1, s_2)$.

mpmathnsum(lambda n1: n1**(-s1) * nsum(lambda n2: n2**(-s2), [1, n1 - 1]), [2, inf])

Typed signatures

Type	Package	Applies
(integer, integer) -> number	analytic	always

Domain: Special functions

Details

• Scoped to depth 2 with both weights $\ge 2$ — every MultiZetaValue identity Fungrim declares is this shape, which is also exactly what makes the double sum converge unconditionally. A general depth-$n$ MZV over compositions is a different, open-ended project and is not attempted here.
• Numerically: $\zeta(s_1,s_2) = \sum_{n\ge1} n^{-s_1} H_{n-1}^{(s_2)}$, summed directly for the first 32 terms; the tail closes with the Euler–Maclaurin expansion of $H_{n-1}^{(s_2)}$ (a Hurwitz-zeta tail), turning each correction term into a plain Riemann-zeta tail from the native Zeta — correctly rounded to a double at every weight Fungrim declares.
• Small cases have closed forms Fungrim states directly: $\zeta(2,2) = \tfrac34\zeta(4)$, $\zeta(3,3) = \tfrac12(\zeta(3)^2-\zeta(6))$, and Euler's reflection $\zeta(a)\zeta(b) - \zeta(a+b) = \zeta(a,b) + \zeta(b,a)$ for $a,b\ge2$.

Examples

6 more kept as data, checked against the oracles

Basic #

$\zeta(2,2) = \tfrac34\zeta(4)$

live ↗

In$N(\mathrm{MultiZetaValue}(2, 2))$Out$0.811\,742\,425\,283\,353\,6$

$\zeta(3,3) = \tfrac12(\zeta(3)^2-\zeta(6))$

live ↗

In$N(\mathrm{MultiZetaValue}(3, 3))$Out$0.213\,798\,868\,224\,592\,5$

Properties #

$\zeta(2, 2) = \tfrac34\zeta(4)$, one of Fungrim's exact closed forms

live ↗

In$\mathrm{chop}(N(\mathrm{MultiZetaValue}(2, 2))-0.75N(ζ(4)))$Out$0.0$

Possible issues #

$s_1 < 2$ stays symbolic — outside the depth-2, both-weights-$\ge2$ scope

live ↗

In$\mathrm{MultiZetaValue}(1, 2)$Out$\mathrm{MultiZetaValue}(1, 2)$

About

Examples	10
Checked against	Wolfram Language 9 agree1 differmpmath 1 unevaluated
Crosswalk	FungrimFungrim entry ×7mpmath

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/multizeta.ts

mappedwolframpackages/oracle/src/mappings.ts

Wolfram has no MultiZetaValue/MZV head; this is the defining double sum, N[]'d since our own evaluator is numeric-only here.

mappedmpmathpackages/oracle/src/mappings.ts

same defining double sum, nested nsum.

See also: Zeta, HurwitzZeta