HurwitzZeta

The Hurwitz zeta function $\zeta(s, a) = \sum_{n=0}^{\infty} (n+a)^{-s}$ for $\operatorname{Re}(s) > 1$, continued analytically elsewhere. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaHurwitz zeta function↗MathWorldHurwitzZetaFunction↗WikidataQ1638777↗DLMF25.11.E1↗FungrimHurwitzZeta✓↗Fungrim entryWolfram LanguageHurwitzZeta!↗

HurwitzZeta(s, a)the Hurwitz zeta function $\zeta(s, a)$.

SageMathhurwitz_zeta(s, a)!↗SymPyzeta(s, a)↗mpmathzeta(s, a)↗

Typed signatures

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

Domain: Special functions

Details

The Hurwitz zeta function $\zeta(s, a) = \sum_{n=0}^{\infty} (n+a)^{-s}$ (DLMF 25.11.1) generalizes Zeta by shifting the summation index by a parameter $a$; the series converges for $\operatorname{Re}(s) > 1$ and $a$ off the nonpositive reals, and continues analytically elsewhere to a function meromorphic in $s$ with a single simple pole at $s = 1$.

• Generalizes the Riemann zeta: $\zeta(s, 1) = \zeta(s)$, and $\zeta(s, m) = \zeta(s) - \sum_{k=1}^{m-1} k^{-s}$ for a positive integer $m$ (so $\zeta(2, 2) = \pi^2/6 - 1$).
• Nonpositive integer $s$: $\zeta(-n, a) = -B_{n+1}(a)/(n+1)$, a Bernoulli polynomial in $a$ -- so $\zeta(0, a) = \tfrac12 - a$ and $\zeta(-1, a) = -\tfrac{1}{12}(6a^2 - 6a + 1)$. See BernoulliB.
• Pole at $s = 1$: $\zeta(1, a) = \text{ComplexInfinity}$ for every $a$.
• Poles at $a = 0, -1, -2, \dots$: the $(n+a)=0$ term is singular. (The two-argument Zeta drops that term instead, staying finite there.)
• Numeric evaluation (under N()) is Euler–Maclaurin summation and supports complex $s$ and $a$; it is aligned with Wolfram's $\mathrm{HurwitzZeta}[s, a]$. Left of $\operatorname{Re}(s) = 0$, where those direct terms cancel, $a$ near the real axis goes through $\zeta(s, 1+h) = \sum_k \binom{-s}{k} h^k \zeta(s+k)$ instead, each $\zeta(s+k)$ from the functional equation.
• That Euler–Maclaurin sum runs in arbitrary precision (as many digits as ce.precision asks for, checked against mpmath at 30 and 50 digits) whenever $s$ and $a$ are both real; a genuinely complex $s$ or $a$ is correct to the same number of digits internally, but the returned value is still one of compute-engine's own complex numbers, which is a pair of doubles -- so asking N() for 30 or 50 digits of a complex result gives back the same ~15-17 correct digits either way, a ceiling of compute-engine's complex-number representation rather than of this computation.
• Certified digits: for exact real $s \ne 1$ and $a > 0$ (integers, rationals or decimals), $\mathrm{N}(\zeta(s, a), d)$ is proven. The Euler–Maclaurin sum runs in ball arithmetic, and its remainder after $M$ Bernoulli pairs is bounded by $4|(s)_{2M}|\,z^{1-s-2M} / ((2\pi)^{2M}(s + 2M - 1))$, $z = N + a$ (Johansson 2015); the d digits are those both ends of the enclosure round to. See N.

Examples

177 more kept as data, checked against the oracles

Basic #

$\zeta(2, 1) = \zeta(2) = \pi^2/6$ -- reduces to the ordinary zeta

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In$\mathrm{HurwitzZeta}(2, 1)$Out$\frac{π^2}{6}$

$\zeta(2, 2) = \pi^2/6 - 1$

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In$\mathrm{HurwitzZeta}(2, 2)$Out$\frac{π^2}{6}-1$

An inexact $a$ evaluates numerically without N(): $\zeta(3, 0.2) = 125.739\ldots$

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In$\mathrm{HurwitzZeta}(3, 0.2)$Out$125.739\,018\,057\,217\,95$

Inside the critical strip, under N()

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In$N(\mathrm{HurwitzZeta}(0.51, 0.87))$Out$-1.320\,155\,023\,694\,958\,375\,51$

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In$\operatorname{Normal}(\operatorname{Series}(\mathrm{HurwitzZeta}(2, x), x, ∞, 6))$Out$\operatorname{Series}(\mathrm{HurwitzZeta}(2, x), x, ∞, 6)$

Scope #

A positive integer $a$ peels off the first terms: $\zeta(7, 5) = \zeta(7) - \sum_{k=1}^{4} k^{-7}$

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In$\mathrm{HurwitzZeta}(7, 5)$Out$ζ(7)-\frac{36\,130\,315}{35\,831\,808}$

Complex $a$

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In$N(\mathrm{HurwitzZeta}(2.3, 8+i))$Out$0.054\,470\,082\,732\,130\,67-0.009\,448\,515\,336\,470\,249i$

Listable: threads over a list of orders

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In$\mathrm{HurwitzZeta}(\bigl\lbrack2, 3, 4\bigr\rbrack, 0.5)$Out$\bigl\lbrack4.934\,802\,200\,544\,679, 8.414\,398\,322\,117\,16, 16.234\,848\,505\,667\,074\bigr\rbrack$

Uncertainty in $a$

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In$\mathrm{HurwitzZeta}(2, \mathrm{Around}(0.5, 0.01))$Out$\mathrm{Around}(4.934\,802\,200\,544\,679, 0.168\,287\,966\,442\,343\,2)$

$\zeta(2, 1/2) = \pi^2/2$ -- the $s = 2$ instance of $\zeta(s, 1/2) = (2^s-1)\zeta(s)$

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In$\mathrm{HurwitzZeta}(2, \frac{1}{2})$Out$\frac{π^2}{2}$

$\zeta(2, 1/4) = \pi^2 + 8G$, with Catalan's constant $G$

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In$\mathrm{HurwitzZeta}(2, \frac{1}{4})$Out$8\operatorname{G}+π^2$

To 30 significant digits, the last one correctly rounded

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In$N(\mathrm{HurwitzZeta}(3, \frac{1}{3}), 30)$Out$27.561\,061\,199\,700\,803\,776\,227\,877\,977\,4$

Far left of the strip, all 40 digits agree with mpmath

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In$N(\mathrm{HurwitzZeta}(-\frac{41}{2}, \frac{3}{10}), 40)$Out$136.361\,935\,791\,831\,144\,171\,775\,099\,936\,398\,861\,428\,6$

1 / 2

Left of the strip at a non-integer $a$

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In$N(\mathrm{HurwitzZeta}(-20, 0.7))$Out$-80.090\,973\,876\,253\,591\,336\,315\,884\,263\,449\,498\,563\,465\,460\,831\,738\,8$

A negative $a$: $(-\tfrac12)^{-5}$ is real, and so is the answer

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In$N(\mathrm{HurwitzZeta}(5, -\frac{1}{2}))$Out$0.144\,760\,409\,444\,467\,716\,272$

Fractional order and shift at fifty digits

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In$N(\mathrm{HurwitzZeta}(\frac{1}{3}, \frac{8}{7}), 50)$Out$-1.138\,936\,744\,449\,099\,174\,654\,867\,433\,453\,572\,781\,096\,191\,946\,075\,5$

Threads over a matrix of shifts, giving Bernoulli-polynomial values

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In$\mathrm{HurwitzZeta}(-1, \bigl\lbrack\bigl\lbrack\frac{1}{2}, 1\bigr\rbrack, \bigl\lbrack1, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{24}, -\frac{1}{12}\bigr\rbrack, \bigl\lbrack-\frac{1}{12}, \frac{1}{24}\bigr\rbrack\bigr\rbrack$

At a negative order it expands to a polynomial in $a$

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In$\mathrm{FunctionExpand}(\mathrm{HurwitzZeta}(-2, a))$Out$-\frac{a^3}{3}+\frac{a^2}{2}-\frac{a}{6}$

At order 0, $\zeta(0,a)=\tfrac12-a$

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In$\mathrm{FunctionExpand}(\mathrm{HurwitzZeta}(0, a))$Out$\frac{1}{2}-a$

Both eigenvalues give the same value, so the matrix is scalar

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In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(()↦\mathrm{HurwitzZeta}(-1, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1/2, 1\bigr\rbrack, \bigl\lbrack1, 1/2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack-\frac{11}{24}, 0\bigr\rbrack, \bigl\lbrack0, -\frac{11}{24}\bigr\rbrack\bigr\rbrack$

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In$\operatorname{Series}(\mathrm{HurwitzZeta}(s, a), a, 0, 3)$Out$\mathrm{HurwitzZeta}(s, 0)-as\mathrm{HurwitzZeta}(s+1, 0)+\frac{1}{2}(s×(s+1)\mathrm{HurwitzZeta}(s+2, 0)a^2)+\frac{1}{6}((-((s+2)\mathrm{HurwitzZeta}(s+3, 0)s^2)-s×(s+2)\mathrm{HurwitzZeta}(s+3, 0))a^3)+O\left(a^4\right)$

Applications #

$\zeta(1/2, 1/3)$ to 40 digits, proven: the Euler–Maclaurin remainder is bounded, and both ends of the enclosure round to these digits

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In$N(\mathrm{HurwitzZeta}(\frac{1}{2}, \frac{1}{3}), 40)$Out$-0.118\,083\,327\,934\,221\,719\,094\,454\,170\,794\,102\,733\,145\,9$

Properties #

$\zeta(0, a) = \tfrac12 - a$ -- exact and symbolic in $a$

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In$\mathrm{HurwitzZeta}(0, a)$Out$\frac{1}{2}-a$

$\zeta(-1, a) = -B_2(a)/2 = -\tfrac{1}{12}(6a^2 - 6a + 1)$

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In$\mathrm{HurwitzZeta}(-1, a)$Out$-\frac{a^2}{2}+\frac{a}{2}-\frac{1}{12}$

Pole at $s = 1$ for every $a$

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In$\mathrm{HurwitzZeta}(1, 3)$Out$\tilde∞$

$\zeta(s, 1/2) = (2^s - 1)\zeta(s)$ for symbolic $s$

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In$\mathrm{HurwitzZeta}(s, \frac{1}{2})$Out$(2^{s}-1)ζ(s)$

$\zeta(s, 1) = \zeta(s)$ for symbolic $s$

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In$\mathrm{HurwitzZeta}(s, 1)$Out$ζ(s)$

$\zeta(0, a) = \tfrac12 - a$ holds at $a = 0$ too

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

$\zeta(-2, a) = -B_3(a)/3$

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In$\mathrm{HurwitzZeta}(-2, a)$Out$-\frac{a^3}{3}+\frac{a^2}{2}-\frac{a}{6}$

$\partial_a\zeta(s, a) = -s\,\zeta(s+1, a)$

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In$\frac{\mathrm{d}\mathrm{HurwitzZeta}(s, a)}{\mathrm{d}a}$Out$-(s\mathrm{HurwitzZeta}(s+1, a))$

$\zeta(4, 2) = \pi^4/90 - 1$

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In$\mathrm{HurwitzZeta}(4, 2)$Out$\frac{π^4}{90}-1$

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

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In$\mathrm{HurwitzZeta}(0, 1)$Out$-\frac{1}{2}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

$\zeta(-1)=-1/12$

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In$\mathrm{HurwitzZeta}(-1, 1)$Out$-\frac{1}{12}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

Shift $1/4$ at order 0

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In$\mathrm{HurwitzZeta}(0, \frac{1}{4})$Out$\frac{1}{4}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

A shifted value at order $-1$

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In$\mathrm{HurwitzZeta}(-1, \frac{1}{4})$Out$\frac{1}{96}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

$\zeta(2)$ minus the first two terms

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In$\mathrm{HurwitzZeta}(2, 3)$Out$\frac{π^2}{6}-\frac{5}{4}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

$\zeta(2)$ minus the first three terms

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In$\mathrm{HurwitzZeta}(2, 4)$Out$\frac{π^2}{6}-\frac{49}{36}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

Shift 5 drops four terms of $\zeta(2)$

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In$\mathrm{HurwitzZeta}(2, 5)$Out$\frac{π^2}{6}-\frac{205}{144}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

Shift 6 drops five terms of $\zeta(2)$

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In$\mathrm{HurwitzZeta}(2, 6)$Out$\frac{π^2}{6}-\frac{5\,269}{3\,600}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

Shift 7 drops six terms of $\zeta(2)$

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In$\mathrm{HurwitzZeta}(2, 7)$Out$\frac{π^2}{6}-\frac{5\,369}{3\,600}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

Shift 8 drops seven terms of $\zeta(2)$

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In$\mathrm{HurwitzZeta}(2, 8)$Out$\frac{π^2}{6}-\frac{266\,681}{176\,400}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

Shift 9 drops eight terms of $\zeta(2)$

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In$\mathrm{HurwitzZeta}(2, 9)$Out$\frac{π^2}{6}-\frac{1\,077\,749}{705\,600}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

Shift 10 drops nine terms of $\zeta(2)$

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In$\mathrm{HurwitzZeta}(2, 10)$Out$\frac{π^2}{6}-\frac{9\,778\,141}{6\,350\,400}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

Reduces to $\zeta(4)=\pi^4/90$

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In$\mathrm{HurwitzZeta}(4, 1)$Out$\frac{π^4}{90}$

Known value · Wolfram MathematicalFunctionData, HurwitzZeta

Possible issues #

$\zeta(-3, 7/3) = -B_4(7/3)/4 = -7813/3240$, exact via the Bernoulli polynomial

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In$\mathrm{HurwitzZeta}(-3, \frac{7}{3})$Out$-\frac{7\,813}{3\,240}$

A pole when the shift is a nonpositive integer

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In$\mathrm{HurwitzZeta}(2, -2)$Out$\tilde∞$

About

Examples	224
Checked against	Wolfram Language 209 agree8 differ2 unevaluatedSageMath 173 agree4 differ6 errorSymPy 208 agree9 differ1 errormpmath 185 agree5 differ6 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×44Wolfram LanguageSageMathSymPympmath

Implementation

referenceenginenotatio

The Dirichlet series, on the principal branch — written exp(−s·ln(n+a)) so the wildcards arrive in the head's own order. It converges only for Re(s) > 1; everywhere else ζ(s, a) is its analytic continuation, which is what the kernel computes.

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/hurwitz-zeta.ts

Euler–Maclaurin in double precision — the fast path, and all a double holds. Asked for more digits than that, N() leaves it: at an integer s ≥ 2 with real a > 0 it takes ζ(n, a) = (−1)ⁿψ⁽ⁿ⁻¹⁾(a)/(n−1)! through compute-engine's PolyGamma, and otherwise (real s ≠ 1, real a > 0) it evaluates the SAME Euler–Maclaurin written as an expression — a finite Sum over Power, Pochhammer and BernoulliB, which compute-engine carries to whatever precision was asked for (packages/symbols/analysis/analytic/src/precise.ts).

compiledgpuwgslpackages/symbols/analysis/analytic/src/shader.ts:zetaWGSL

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

mappedwolframpackages/oracle/src/mappings.ts

mpmath is correct at negative-integer s where Wolfram's N[] is not; the grid in special-functions.examples.json covers that branch.

mappedsympypackages/oracle/src/mappings.ts

mpmath is correct at negative-integer s where Wolfram's N[] is not; the grid in special-functions.examples.json covers that branch.

mappedmpmathpackages/oracle/src/mappings.ts

mpmath is correct at negative-integer s where Wolfram's N[] is not; the grid in special-functions.examples.json covers that branch.

mappedsagepackages/oracle/src/mappings.ts

mpmath is correct at negative-integer s where Wolfram's N[] is not; the grid in special-functions.examples.json covers that branch.

See also: Zeta, BernoulliB, Gamma, Digamma