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Zeta ​

The Riemann zeta function for , continued analytically elsewhere.

WikidataQ187235Encyclopedia of MathematicsZeta-function#Riemann.27s_zeta-functionBritannicatopic/Riemann-zeta-functionFungrimRiemannZeta✓Fungrim entryWolfram LanguageZeta✓
Zeta(s)the Riemann zeta function .
WikipediaRiemann zeta functionMathWorldRiemannZetaFunctionDLMF25.2.E1SageMathzeta(s)SymPyzeta(s)mpmathzeta(s)
Zeta(s, a)the two-argument generalized (Hurwitz-type) zeta ; see HurwitzZeta for the difference.
WikipediaHurwitz zeta functionMathWorldHurwitzZetaFunctionDLMF25.11.E1FungrimHurwitzZetaSageMathhurwitz_zeta(s, a)SymPyzeta(s, a)mpmathzeta(s, a)

Domain: Special functions

Details
  • Even positive integers have closed forms in powers of : (the Basel problem), , and so on.
  • Trivial zeros at the negative even integers: .
  • Special values at nonpositive integers relate to the Bernoulli numbers: . See BernoulliB.
  • Pole at : , reflecting the divergence of the harmonic series.
  • Odd integers (Apéry's constant , and beyond) have no known closed form and stay numeric-only.
  • compute-engine reduces Zeta to an exact closed form at these special integer points -- even under plain evaluation, and even threaded over a list -- but leaves other reals, like , symbolic pending N().
  • The two-argument form is provided by `@enumeratio/analytic` (compute-engine's built-in Zeta is single-argument). It equals HurwitzZeta for but follows Wolfram's generalized-zeta convention for : the terms off the positive axis use and the term is dropped, so stays finite at -- in particular -- where HurwitzZeta has poles.

Examples

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/analytic/src/hurwitz-zeta.ts

Riemann ζ is the m = 1 case of the Hurwitz implementation.

compiledgpuwgslpackages/analytic/src/shader.ts:zetaWGSL

Produces a colour per pixel — complex ζ, domain-coloured.

compute-engine's WGSLTarget has no lowering for Zeta (it fails closed), so the real-scalar GPU path falls back to CPU. `zetaWGSL` is used only by the complex-portrait fragment shader (complex-plot.ts), where Zeta(s) becomes clogPolar(hurwitz(s, vec2f(1,0))).

componentbrowser<notatio-complex-plot>packages/notatio-lit/src/notatio-complex-plot.ts

Produces a rendered phase portrait.

For a plotted ζ the head's meaning bottoms out here: the element IS the answer, and there is no expression it reduces to.

mappedexternalwolfram / sympy / mpmath / sagepackages/oracle/src/mappings.ts

Keyed by signature, not name: one argument is Riemann's, two is Hurwitz's, and the systems disagree about how to spell that.

See also: HurwitzZeta, BernoulliB, Gamma, Digamma