Zeta

The Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ for $\operatorname{Re}(s) > 1$, continued analytically elsewhere.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikidataQ187235↗Encyclopedia of MathematicsZeta-function#Riemann.27s_zeta-function↗Britannicatopic/Riemann-zeta-function↗FungrimRiemannZeta✓↗Fungrim entryWolfram LanguageZeta!↗

Zeta(s)the Riemann zeta function $\zeta(s)$.

WikipediaRiemann zeta function↗MathWorldRiemannZetaFunction↗DLMF25.2.E1↗SageMathzeta(s)!↗SymPyzeta(s)↗mpmathzeta(s)↗Juliaenumeratio_ball(zeta, s)

Zeta(s, a)the two-argument generalized (Hurwitz-type) zeta $\zeta(s, a)$; see HurwitzZeta for the difference.

WikipediaHurwitz zeta function↗MathWorldHurwitzZetaFunction↗DLMF25.11.E1↗FungrimHurwitzZeta↗SageMathhurwitz_zeta(s, a)!↗SymPyzeta(s, a)↗mpmathzeta(s, a)↗Juliaenumeratio_ball(zeta, s, a)

Typed signatures

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

Domain: Special functions

Details

The Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$, for $\operatorname{Re}(s) > 1$, extends by analytic continuation to a meromorphic function on all of $\mathbb{C}$ with a single simple pole at $s = 1$. Over the primes it factors as the Euler product $\zeta(s) = \prod_p (1 - p^{-s})^{-1}$ (DLMF 25.2.11), which is where its role in the distribution of primes comes from. It is HurwitzZeta's $a = 1$ slice, $\zeta(s) = \zeta(s, 1)$, and PolyLog's $z = 1$ edge, $\zeta(s) = \operatorname{Li}_s(1)$.

• Even positive integers have closed forms in powers of $\pi$: $\zeta(2) = \pi^2/6$ (the Basel problem), $\zeta(4) = \pi^4/90$, and so on.
• Trivial zeros at the negative even integers: $\zeta(-2n) = 0$.
• Special values at nonpositive integers relate to the Bernoulli numbers: $\zeta(-n) = -B_{n+1}/(n+1)$. See BernoulliB.
• Pole at $s=1$: $\zeta(1) = \text{ComplexInfinity}$, reflecting the divergence of the harmonic series.
• Odd integers $\geq 3$ (Apéry's constant $\zeta(3)$, 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 $\zeta(3)$, symbolic pending N().
• The two-argument form $\zeta(s, a) = \sum_{n\geq 0} (n+a)^{-s}$ is provided by @enumeratio/analytic (compute-engine's built-in Zeta is single-argument). It equals HurwitzZeta for $\operatorname{Re}(a) > 0$ but follows Wolfram's generalized-zeta convention for $a \leq 0$: the terms off the positive axis use $((n+a)^2)^{-s/2}$ and the $n+a=0$ term is dropped, so $\zeta(s, a)$ stays finite at $a = 0, -1, -2, \dots$ -- in particular $\zeta(s, 0) = \zeta(s)$ -- where HurwitzZeta has poles.
• Certified digits: at an exact real $s \ne 1$ (an integer, a rational or a decimal), $\mathrm{N}(\zeta(s), d)$ is proven, not just agreed on, and so is $\mathrm{N}(\zeta(s, a), d)$ for an exact $a > 0$. See HurwitzZeta for the bound, and N.

Plot

ComplexPlot

real slice

Examples

25 more kept as data, checked against the oracles

Basic #

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In$ζ(2)$Out$\frac{π^2}{6}$

Known value · DLMF 25.6.1

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In$ζ(4)$Out$\frac{π^4}{90}$

Known value · DLMF 25.6.1

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In$ζ(6)$Out$\frac{π^6}{945}$

Known value · DLMF 25.6.1

Threads over a list, reducing every element to its exact closed form

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In$ζ(\bigl\lbrack-1, -2, -3\bigr\rbrack)$Out$\bigl\lbrack-\frac{1}{12}, 0, \frac{1}{120}\bigr\rbrack$

Apéry's constant $\zeta(3)$, numerically

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In$N(ζ(3))$Out$1.202\,056\,903\,159\,594\,285\,4$

A machine-precision argument

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In$ζ(1.5)$Out$2.612\,375\,348\,685\,488\,343\,35$

$\zeta(8) = \pi^8/9450$

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In$ζ(8)$Out$\frac{π^8}{9\,450}$

$\zeta(-5) = -B_6/6 = -1/252$

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In$ζ(-5)$Out$-\frac{1}{252}$

Zeta(s, -n) = ζ(s) + Σ j^(-s): the terms left of the dropped n+a=0 slot count as |n+a|^(-s).

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In$ζ(2, -1)$Out$1+\frac{π^2}{6}$

Known value · Wolfram Zeta[s, a], checked with wolframscript: ζ(s) + Σ_{j=1}^{n} j^(-s) (the n+a=0 term dropped); DLMF 25.11.1 defines the sum only for a ≠ 0, −1, −2, …

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In$ζ(3, -1)$Out$1+ζ(3)$

Known value · Wolfram Zeta[s, a], checked with wolframscript: ζ(s) + Σ_{j=1}^{n} j^(-s) (the n+a=0 term dropped); DLMF 25.11.1 defines the sum only for a ≠ 0, −1, −2, …

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In$ζ(3, -2)$Out$\frac{9}{8}+ζ(3)$

Known value · Wolfram Zeta[s, a], checked with wolframscript: ζ(s) + Σ_{j=1}^{n} j^(-s) (the n+a=0 term dropped); DLMF 25.11.1 defines the sum only for a ≠ 0, −1, −2, …

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In$ζ(4, -3)$Out$\frac{1\,393}{1\,296}+\frac{π^4}{90}$

Known value · Wolfram Zeta[s, a], checked with wolframscript: ζ(s) + Σ_{j=1}^{n} j^(-s) (the n+a=0 term dropped); DLMF 25.11.1 defines the sum only for a ≠ 0, −1, −2, …

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In$N(ζ(3, -3))$Out$2.364\,093\,940\,196\,631\,322\,44$

Known value · Wolfram Zeta[s, a], checked with wolframscript: ζ(s) + Σ_{j=1}^{n} j^(-s) (the n+a=0 term dropped); DLMF 25.11.1 defines the sum only for a ≠ 0, −1, −2, …

The convention holds left of the strip too: ζ(−1, −1) = ζ(−1) + 1, not the Bernoulli-polynomial −13/12 DLMF 25.11.14 gives where the series is meant.

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In$ζ(-1, -1)$Out$\frac{11}{12}$

Known value · Wolfram Zeta[s, a], checked with wolframscript: ζ(s) + Σ_{j=1}^{n} j^(-s) (the n+a=0 term dropped); DLMF 25.11.1 defines the sum only for a ≠ 0, −1, −2, …

At n = 1 any exact $s$ gives $1 + \zeta(s)$; a longer sum stays symbolic unless $\zeta(s)$ has a closed form.

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In$ζ(\frac{5}{2}, -1)$Out$1+ζ(\frac{5}{2})$

Known value · Wolfram Zeta[s, a], checked with wolframscript: ζ(s) + Σ_{j=1}^{n} j^(-s) (the n+a=0 term dropped); DLMF 25.11.1 defines the sum only for a ≠ 0, −1, −2, …

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In$\mathrm{FullSimplify}(\operatorname{Normal}(\operatorname{Series}(ζ(x), x, 0, 2)))$Out$\frac{1}{2}\Zeta^{\prime\prime}(0)x^2+x\Zeta^{\prime}(0)-\frac{1}{2}$

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In$\operatorname{Normal}(\operatorname{Series}(ζ(x), x, ∞, 6))$Out$\operatorname{Series}(ζ(x), x, ∞, 6)$

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In$\operatorname{Normal}(\operatorname{Series}(ζ(x), x, 1, 2))$Out$\frac{1}{2}\mathrm{StieltjesGamma}(2)(x-1)^2-(x-1)\mathrm{StieltjesGamma}(1)+\frac{1}{x-1}+γ$

Scope #

On the critical line, $\zeta(1/2 + 14i) = 0.02224\ldots - 0.10326\ldots i$ — complex $s$, from @enumeratio/analytic (compute-engine's Zeta evaluates real $s$ only). Computed in arbitrary precision and rounded once, so each part is the nearest double in every JavaScript engine

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In$N(ζ(0.5+14i))$Out$0.022\,241\,142\,609\,993\,59-0.103\,258\,123\,266\,450\,06i$

Two-argument $\zeta(s, a)$ (via @enumeratio/analytic): $\zeta(s, 1) = \zeta(s)$ recovers the ordinary case, so $\zeta(2, 1) = \pi^2/6$

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In$ζ(2, 1)$Out$\frac{π^2}{6}$

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

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In$ζ(2, 2)$Out$\frac{π^2}{6}-1$

$\zeta(s, 0) = \zeta(s)$: the $n+a=0$ term is dropped, so unlike HurwitzZeta there is no pole at $a = 0$

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In$ζ(2, 0)$Out$\frac{π^2}{6}$

Arbitrary real arguments

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In$ζ(5.211\,111\,111\,111\,111)$Out$1.031\,391\,796\,934\,384\,998\,39$

Left of the critical strip, via the functional equation

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In$ζ(-4.5)$Out$-0.003\,091\,669\,247\,215\,833\,844\,82$

Complex $s$ under N()

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In$N(ζ(2-5i))$Out$0.850\,962\,943\,624\,262\,8-0.098\,996\,946\,134\,831\,25i$

$\zeta(\infty) = 1$: only the $n = 1$ term survives

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In$ζ(∞)$Out$1$

Two arguments at a nonpositive integer $s$: $\zeta(-1, 5) = -B_2(5)/2 = -121/12$

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In$ζ(-1, 5)$Out$-\frac{121}{12}$

Two arguments, inexact

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In$ζ(-1.5, 5)$Out$-17.050\,064\,749\,342\,655\,014\,129\,5$

Two complex arguments under N(), to ten digits

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In$N(ζ(-1.5+i, 2.5-i), 10)$Out$0.018\,486\,809\,19+1.675\,533\,784i$

...and with inexact complex arguments, without N(), as the real case does

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In$ζ(-1.5+i, 2.5-i)$Out$0.018\,486\,809\,193\,810\,65+1.675\,533\,784\,301\,766i$

$\zeta(20, 1/2) = (2^{20} - 1)\zeta(20)$, an exact multiple of $\pi^{20}$

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In$ζ(20, \frac{1}{2})$Out$\frac{221\,930\,581π^{20}}{1\,856\,156\,927\,625}$

$\zeta(2, 1/2) = 3\zeta(2) = \pi^2/2$

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

Interval arithmetic: $\zeta$ is decreasing on $(1, \infty)$

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In$ζ(\mathrm{Interval}(1.1, 1.2))$Out$\mathrm{Interval}(5.591\,582\,441\,177\,75, 10.584\,448\,464\,950\,812)$

Uncertainty in $a$, propagated with $\partial_a\zeta(s, a) = -s\,\zeta(s+1, a)$

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In$ζ(\frac{1}{2}, \mathrm{Around}(0.5, 0.01))$Out$\mathrm{Around}(-0.604\,898\,643\,421\,630\,4, 0.023\,882\,689\,737\,774\,167)$

To 40 significant digits, every one correctly rounded

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In$N(ζ(3), 40)$Out$1.202\,056\,903\,159\,594\,285\,399\,738\,161\,511\,449\,990\,765$

Fifty digits just right of the pole at $s = 1$

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In$N(ζ(\frac{5}{4}), 50)$Out$4.595\,111\,825\,842\,943\,380\,685\,378\,039\,694\,625\,652\,281\,029\,780\,604\,8$

$\zeta(3,2) = \zeta(3) - 1$, to 50 digits

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In$N(ζ(3, 2), 50)$Out$0.202\,056\,903\,159\,594\,285\,399\,738\,161\,511\,449\,990\,764\,986\,292\,340\,5$

Threads over a matrix, with closed forms at half-integer shifts

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In$ζ(2, \bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{3}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{π^2}{2}, \frac{π^2}{6}\bigr\rbrack, \bigl\lbrack\frac{π^2}{6}, ζ(2, \frac{3}{2})\bigr\rbrack\bigr\rbrack$

$\zeta(4,5) = \frac{\pi^4}{90} - \sum_{k=1}^{4} k^{-4}$

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In$ζ(4, 5)$Out$\frac{π^4}{90}-\frac{22\,369}{20\,736}$

A pole at 1, closed forms at 2 and 4, and $\zeta(3)$ stays symbolic

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In$ζ(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\bigl\lbrack\tilde∞, \frac{π^2}{6}, ζ(3), \frac{π^4}{90}\bigr\rbrack$

MatrixFunction on a diagonal matrix matches entrywise Hurwitz values

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦ζ(2, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{3}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{π^2}{2}, 0\bigr\rbrack, \bigl\lbrack0, ζ(2, \frac{3}{2})\bigr\rbrack\bigr\rbrack$

$\zeta(\tfrac12+14i)$, both parts correctly rounded: the bignum kernel rather than a double one, which was off in the 14th digit

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In$N(ζ(0.5+14i))$Out$0.022\,241\,142\,609\,993\,59-0.103\,258\,123\,266\,450\,06i$

Applications #

The probability that two random positive integers are coprime is $1/\zeta(2) = 6/\pi^2$

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In$\frac{1}{ζ(2)}=\frac{6}{π^2}$Out$\mathrm{True}$

Apéry's constant $\zeta(3)$ to 40 digits, proven: both ends of the enclosure round to these digits

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In$N(ζ(3), 40)$Out$1.202\,056\,903\,159\,594\,285\,399\,738\,161\,511\,449\,990\,765$

Properties #

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

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In$ζ(0)$Out$-\frac{1}{2}$

Known value · DLMF 25.6.1

Trivial zero: $\zeta(-2n) = 0$ for positive integer n

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In$ζ(-2)$Out$0$

$\zeta(-3) = -B_4/4 = 1/120$, via the Bernoulli-number relation. See BernoulliB

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In$ζ(-3)$Out$\frac{1}{120}$

Pole at $s=1$: the harmonic series $\sum 1/n$ diverges

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In$ζ(1)$Out$\tilde∞$

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

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In$ζ(s, 1)$Out$ζ(s)$

$\zeta(0) = -\frac12$, the regularised value

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In$ζ(0, 1)$Out$-\frac{1}{2}$

Known value · Wolfram MathematicalFunctionData, Zeta

The famous $\zeta(-1) = -\frac1{12}$

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In$ζ(-1, 1)$Out$-\frac{1}{12}$

Known value · Wolfram MathematicalFunctionData, Zeta

$\zeta(0, a) = \frac12 - a$, here $\frac14$ at $a = \frac14$

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In$ζ(0, \frac{1}{4})$Out$\frac{1}{4}$

Known value · Wolfram MathematicalFunctionData, Zeta

A negative-order value via Bernoulli polynomials: $\frac1{96}$

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

Known value · Wolfram MathematicalFunctionData, Zeta

$\zeta(2,3) = \frac{\pi^2}{6} - 1 - \frac14$

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

Known value · Wolfram MathematicalFunctionData, Zeta

Subtract the first three terms of the Basel series

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

Known value · Wolfram MathematicalFunctionData, Zeta

Remove $1 + \frac14 + \frac19 + \frac1{16}$ from $\pi^2/6$

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

Known value · Wolfram MathematicalFunctionData, Zeta

The same pattern with denominator 3600

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

Known value · Wolfram MathematicalFunctionData, Zeta

A shift of 7 changes only the rational part

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

Known value · Wolfram MathematicalFunctionData, Zeta

Rational offset grows with the shift

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

Known value · Wolfram MathematicalFunctionData, Zeta

Another shift of the Basel tail

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

Known value · Wolfram MathematicalFunctionData, Zeta

Rational part of the tail from 10

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

Known value · Wolfram MathematicalFunctionData, Zeta

At $a = 1$ this is the plain $\zeta(4)=\pi^4/90$

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In$ζ(4, 1)$Out$\frac{π^4}{90}$

Known value · Wolfram MathematicalFunctionData, Zeta

The tail from 2 subtracts exactly 1

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In$ζ(4, 2)$Out$\frac{π^4}{90}-1$

Known value · Wolfram MathematicalFunctionData, Zeta

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In$\sum_{k=1}^1ζ(2k)ζ(4-2k)$Out$\frac{π^4}{36}$

Known value · Wolfram MathematicalFunctionData, Zeta:Riemann, FunctionalEquations

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In$\sum_{k=1}^2ζ(2k)ζ(6-2k)$Out$\frac{π^6}{270}$

Known value · Wolfram MathematicalFunctionData, Zeta:Riemann, FunctionalEquations

$\zeta(0, a) = \frac12 - a$ holds at $a = 0$ too: at $s = 0$ the $n+a=0$ term is $0^0 = 1$, kept

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In$ζ(0, 0)$Out$\frac{1}{2}$

Possible issues #

Odd integers $\geq 3$ have no known closed form (Apéry's constant, $\zeta(3)$), so compute-engine leaves them symbolic under plain evaluation -- use N() for a decimal

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In$ζ(3)$Out$ζ(3)$

$\zeta(3, -1/2) = 8 + \zeta(3, 1/2)$ under Wolfram's generalized convention for $a \leq 0$ — differs from HurwitzZeta there

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In$N(ζ(3, -\frac{1}{2}))$Out$16.414\,398\,322\,117\,159\,997\,8$

$\zeta(-1, -2) = 35/12$ exactly, but at a negative $a$ this stays symbolic without N()

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In$N(ζ(-1, -2))$Out$2.916\,666\,666\,666\,666\,666\,67$

Neat examples #

The notorious "sum of all positive integers" result from zeta-function regularization, $\zeta(-1) = -1/12$

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In$ζ(-1)$Out$-\frac{1}{12}$

About

Examples	95
Checked against	Wolfram Language 86 agree6 differ1 unevaluatedSageMath 60 agree12 differ7 unevaluated13 errorSymPy 61 agree15 differ17 unevaluated1 errormpmath 53 agree12 differ3 unevaluated17 errorJulia 52 agree11 differ3 unevaluated19 error
Crosswalk	Wikipedia ×2MathWorld ×2WikidataEncyclopedia of MathematicsBritannicaDLMF ×2Fungrim ×2Fungrim entry ×48Wolfram LanguageSageMath ×2SymPy ×2mpmath ×2Julia ×2

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

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

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

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

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

compute-engine's WGSLTarget has no lowering for Zeta; ours compiles Zeta(s) as zetaGen(s, 1) on the real-scalar path, and the complex-portrait fragment shader (complex-plot.ts) takes it as clogPolar(hurwitz(s, vec2f(1,0))).

componentbrowser<notatio-complex-plot>packages/components/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.

mappedwolframpackages/oracle/src/mappings.ts

Riemann ζ at one argument, in every system.

mappedsympypackages/oracle/src/mappings.ts

Riemann ζ at one argument, in every system.

mappedmpmathpackages/oracle/src/mappings.ts

Riemann ζ at one argument, in every system.

mappedsagepackages/oracle/src/mappings.ts

Riemann ζ at one argument, in every system.

mappedwolframpackages/oracle/src/mappings.ts

Two arguments is Hurwitz. Wolfram's Zeta[s, a] drops the n+a=0 term like ours does (its HurwitzZeta[s, 0] diverges); Sage spells it hurwitz_zeta; SymPy and mpmath overload zeta.

mappedsympypackages/oracle/src/mappings.ts

Two arguments is Hurwitz. Wolfram's Zeta[s, a] drops the n+a=0 term like ours does (its HurwitzZeta[s, 0] diverges); Sage spells it hurwitz_zeta; SymPy and mpmath overload zeta.

mappedmpmathpackages/oracle/src/mappings.ts

Two arguments is Hurwitz. Wolfram's Zeta[s, a] drops the n+a=0 term like ours does (its HurwitzZeta[s, 0] diverges); Sage spells it hurwitz_zeta; SymPy and mpmath overload zeta.

mappedsagepackages/oracle/src/mappings.ts

Two arguments is Hurwitz. Wolfram's Zeta[s, a] drops the n+a=0 term like ours does (its HurwitzZeta[s, 0] diverges); Sage spells it hurwitz_zeta; SymPy and mpmath overload zeta.

mappedjuliapackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: HurwitzZeta, BernoulliB, Gamma, Digamma, RiemannSiegelZ, RiemannZetaZero