DirichletBeta

The Dirichlet beta function $\beta(s) = \sum_{n\ge0} (-1)^n (2n+1)^{-s}$, the L-function of the nontrivial character mod 4 — entire, with $\beta(1) = \pi/4$ and $\beta(2) = G$ (Catalan).

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaDirichlet beta function↗MathWorldDirichletBetaFunction↗WikidataQ1227706↗Wolfram LanguageDirichletBeta!↗

DirichletBeta(s)the Dirichlet beta function $\beta(s)$.

mpmathdirichlet(s, [0, 1, 0, -1])Juliaenumeratio_ball(s -> (polylog(s, onei(CC)) - polylog(s, -onei(CC))) / (2 * onei(CC)), s)

Typed signatures

Type	Package	Applies
(complex | infinity) -> number	analyticreplaces compute-engine	always
(complex | infinity) -> number	compute-engine	always

Domain: Special functions

Details

• $\beta(s) = 4^{-s}\left(\zeta(s, \tfrac14) - \zeta(s, \tfrac34)\right)$ in terms of HurwitzZeta, which is how it is evaluated numerically (complex $s$ included).
• Odd positive integers have closed forms in $\pi$ and the Euler numbers: $\beta(2k+1) = (-1)^k E_{2k}\,\pi^{2k+1} / (4^{k+1}(2k)!)$ — $\beta(1) = \pi/4$ (Leibniz), $\beta(3) = \pi^3/32$, $\beta(5) = 5\pi^5/1536$.
• Even positive integers do not: $\beta(2) = G = 0.9159\ldots$ is Catalan's constant, and $\beta(4), \beta(6), \dots$ stay symbolic.
• Nonpositive integers are Euler numbers: $\beta(-2k) = E_{2k}/2$ ($\beta(0) = \tfrac12$, $\beta(-2) = -\tfrac12$, $\beta(-4) = \tfrac52$), and $\beta(-(2k+1)) = 0$.
• Functional equation $\beta(1-s) = (\pi/2)^{-s} \sin(\pi s/2)\,\Gamma(s)\,\beta(s)$; entire, no poles.

Plot

ComplexPlot

real slice

Examples

2 more kept as data, checked against the oracles

Basic #

$\beta(1) = \pi/4$: Leibniz's series $1 - \tfrac13 + \tfrac15 - \cdots$

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In$\mathrm{DirichletBeta}(1)$Out$\frac{π}{4}$

$\beta(2) = G$, Catalan's constant

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In$\mathrm{DirichletBeta}(2)$Out$\operatorname{G}$

$\beta(3) = \pi^3/32$

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In$\mathrm{DirichletBeta}(3)$Out$\frac{π^3}{32}$

Numeric for a floating-point argument — through $4^{-s}(\zeta(s,\tfrac14) - \zeta(s,\tfrac34))$, so it answers to the engine's working precision rather than a double's

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In$\mathrm{DirichletBeta}(2.5)$Out$0.948\,622\,174\,037\,054\,707\,445$

A symbolic argument stays symbolic

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In$\mathrm{DirichletBeta}(s)$Out$\mathrm{DirichletBeta}(s)$

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In$\mathrm{Map}(\mathrm{DirichletBeta}, \bigl\lbrack2, 3\bigr\rbrack)$Out$\bigl\lbrack\operatorname{G}, \frac{π^3}{32}\bigr\rbrack$

Scope #

Complex arguments, here on the critical line

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In$\mathrm{DirichletBeta}(0.5+14i)$Out$1.537\,115\,438\,440\,316+1.343\,451\,426\,867\,757\,3i$

Threads elementwise over an array

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In$\mathrm{DirichletBeta}(\bigl\lbrack\bigl\lbrack1, -1\bigr\rbrack, \bigl\lbrack-1, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{π}{4}, 0\bigr\rbrack, \bigl\lbrack0, \frac{π}{4}\bigr\rbrack\bigr\rbrack$

$\beta(s) \to 1$ as $s \to \infty$

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In$\mathrm{DirichletBeta}(∞)$Out$1$

Uncertainty propagation: $\beta(2 \pm 0.01) = G \pm 0.00082$

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In$\mathrm{DirichletBeta}(\mathrm{Around}(2, 0.01))$Out$\mathrm{Around}(0.915\,965\,594\,177\,219, 0.000\,815\,807\,361\,165\,928)$

To exactly 40 significant digits, as many as asked for

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In$N(\mathrm{DirichletBeta}(2), 40)$Out$0.915\,965\,594\,177\,219\,015\,054\,603\,514\,932\,384\,110\,774\,1$

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In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\mathrm{DirichletBeta}, \bigl\lbrack\bigl\lbrack1, -1\bigr\rbrack, \bigl\lbrack-1, 1\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack\frac{1}{4}+\frac{\operatorname{G}}{2}, \frac{1}{4}-\frac{\operatorname{G}}{2}\bigr\rbrack, \bigl\lbrack\frac{1}{4}-\frac{\operatorname{G}}{2}, \frac{1}{4}+\frac{\operatorname{G}}{2}\bigr\rbrack\bigr\rbrack$

Properties #

$\beta(7) = 61\pi^7/184320$ — the $61$ is the Euler number $|E_6|$

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In$\mathrm{DirichletBeta}(7)$Out$\frac{61π^7}{184\,320}$

$\beta(-2k) = E_{2k}/2$ and $\beta(-(2k+1)) = 0$

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In$\mathrm{DirichletBeta}(\bigl\lbrack0, -1, -2, -3, -4\bigr\rbrack)$Out$\bigl\lbrack\frac{1}{2}, 0, -\frac{1}{2}, 0, \frac{5}{2}\bigr\rbrack$

$\beta(5) = 5\pi^5/1536$

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In$\mathrm{DirichletBeta}(5)$Out$\frac{5π^5}{1\,536}$

Zero at negative odd integers

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In$\mathrm{DirichletBeta}(-5)$Out$0$

Known value · Wolfram MathematicalFunctionData, DirichletBeta

$\beta(-4) = \frac52$, an Euler number over 2

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In$\mathrm{DirichletBeta}(-4)$Out$\frac{5}{2}$

Known value · Wolfram MathematicalFunctionData, DirichletBeta

$\beta(-3) = 0$, a trivial zero

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In$\mathrm{DirichletBeta}(-3)$Out$0$

Known value · Wolfram MathematicalFunctionData, DirichletBeta

$\beta(-2) = -\frac12$

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In$\mathrm{DirichletBeta}(-2)$Out$-\frac{1}{2}$

Known value · Wolfram MathematicalFunctionData, DirichletBeta

The first trivial zero, $\beta(-1) = 0$

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In$\mathrm{DirichletBeta}(-1)$Out$0$

Known value · Wolfram MathematicalFunctionData, DirichletBeta

$\beta(0) = \frac12$

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In$\mathrm{DirichletBeta}(0)$Out$\frac{1}{2}$

Known value · Wolfram MathematicalFunctionData, DirichletBeta

Possible issues #

Even $s \ge 4$ has no closed form and stays symbolic; N() gives $0.98894\ldots$

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In$\mathrm{DirichletBeta}(4)$Out$\mathrm{DirichletBeta}(4)$

About

Examples	24
Checked against	Wolfram Language 21 agree3 differSymPy 20 agree1 differ1 unevaluatedmpmath 19 agree1 differJulia 19 agree1 error
Crosswalk	WikipediaMathWorldWikidataWolfram LanguagempmathJulia

Implementation

referenceenginenotatio

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/dirichlet.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

DirichletBeta[s]; mpmath.dirichlet(s, [0, 1, 0, -1]).

mappedmpmathpackages/oracle/src/mappings.ts

The Dirichlet L-function of the character mod 4.

mappedjuliapackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy has no Dirichlet beta; this is its polylog definition β(s) = (Li_s(i) − Li_s(−i)) / 2i.

See also: DirichletEta, HurwitzZeta, ClausenCl, Zeta