# Zeta and L-functions Part of the [Fungrim Identities](/compute-engine/reference/fungrim/) reference — **80 identities** for zeta and l-functions. :::info[Generated reference] This page is generated from the compiled Fungrim artifact by `scripts/fungrim/gen-reference-doc.ts` (upstream snapshot `3a299164c683`, translator `grim2mathjson 0.1.0`). Do not edit it by hand. The corpus is MIT-licensed; see `data/fungrim/LICENSE`. ::: ## Contents - [Dirichlet characters](#dirichlet-characters) (15) - [Hurwitz zeta function](#hurwitz-zeta-function) (41) - [Multiple zeta values](#multiple-zeta-values) (7) - [Riemann zeta function](#riemann-zeta-function) (17) ## Dirichlet characters $$\mathrm{DirichletL}(1, \mathrm{DirichletCharacter}(4, 3))=\frac{\pi}{4}$$ **Symbols:** **DirichletCharacter** — Dirichlet character; **DirichletL** — Dirichlet L-function. Used by the Compute Engine for simplification. [`3b8c97` · Fungrim entry ↗](https://fungrim.org/entry/3b8c97) --- $$\mathrm{DirichletGroup}(q)=\mathrm{Map}(\mathrm{Filter}(1..\max(q, 2)-1, \ell\mapsto\gcd(\ell, q)=1), \ell\mapsto\mathrm{DirichletCharacter}(q, \ell))$$ **Holds when** $q\in\N^*$. **Symbols:** **DirichletCharacter** — Dirichlet character; **DirichletGroup** — Dirichlet characters with given modulus. Used by the Compute Engine for simplification. [`47d430` · Fungrim entry ↗](https://fungrim.org/entry/47d430) --- $$\mathrm{DirichletCharacter}(p^{e_{var}}, \ell, n)=\exp(\frac{2\pi\imaginaryI\mathrm{DiscreteLog}(\ell, \mathrm{ConreyGenerator}(p), p^{e_{var}})\mathrm{DiscreteLog}(n, \mathrm{ConreyGenerator}(p), p^{e_{var}})}{\mathrm{Totient}(p^{e_{var}})})$$ **Holds when** $p\in\mathrm{Primes}\land p\ge3\land e_{var}\in\N^*\land\ell\in1..p^{e_{var}}-1\land n\in\Z\land\gcd(\ell, p^{e_{var}})=\gcd(n, p^{e_{var}})=1$. **Symbols:** **ConreyGenerator** — Conrey generator; **DirichletCharacter** — Dirichlet character; **DiscreteLog** — Discrete logarithm. Used by the Compute Engine for simplification. [`4cf4e4` · Fungrim entry ↗](https://fungrim.org/entry/4cf4e4) --- $$\mathrm{ConreyGenerator}(p)=\begin{cases}10&p=40\,487\\7&p=6\,692\,367\,337\\\min(\mathrm{Filter}(\N^*, a\mapsto\mathrm{Count}(\mathrm{Map}(\N, k\mapsto a^{k}\bmod p))=p-1))&\top\end{cases}$$ **Holds when** $p\in\mathrm{Primes}\land p\ge3\land p\lt10^{12}$. **Symbols:** **ConreyGenerator** — Conrey generator. Used by the Compute Engine for simplification. [`540931` · Fungrim entry ↗](https://fungrim.org/entry/540931) --- $$\mathrm{Count}(\mathrm{DirichletGroup}(q))=\mathrm{Totient}(q)$$ **Holds when** $q\in\N^*$. **Symbols:** **DirichletGroup** — Dirichlet characters with given modulus. Used by the Compute Engine for simplification. [`62f7d5` · Fungrim entry ↗](https://fungrim.org/entry/62f7d5) --- $$\mathrm{ConreyGenerator}(p)=\min(\mathrm{Filter}(\N^*, a\mapsto\mathrm{Count}(\mathrm{Map}(\N, k\mapsto a^{k}\bmod p))=p-1\land\mathrm{Count}(\mathrm{Map}(\N, k\mapsto a^{k}\bmod p^2))=p(p-1)))$$ **Holds when** $p\in\mathrm{Primes}\land p\ge3$. **Symbols:** **ConreyGenerator** — Conrey generator. Used by the Compute Engine for simplification. [`75231e` · Fungrim entry ↗](https://fungrim.org/entry/75231e) --- $$\mathrm{DirichletL}(0, \mathrm{DirichletCharacter}(q, 1))=\begin{cases}-(\frac{1}{2})&q=1\\0&\top\end{cases}$$ **Holds when** $q\in\N^*$. **Symbols:** **DirichletCharacter** — Dirichlet character; **DirichletL** — Dirichlet L-function. Used by the Compute Engine for simplification. [`a07d28` · Fungrim entry ↗](https://fungrim.org/entry/a07d28) --- $$\mathrm{DirichletL}(s, \mathrm{DirichletCharacter}(1, 1))=\Zeta(s)$$ **Holds when** $s\in\C$. **Symbols:** **DirichletCharacter** — Dirichlet character; **DirichletL** — Dirichlet L-function. Used by the Compute Engine for simplification. [`a9337b` · Fungrim entry ↗](https://fungrim.org/entry/a9337b) --- $$\mathrm{DirichletL}(1, \mathrm{DirichletCharacter}(5, 4))=\frac{2\ln(\varphi)}{\sqrt{5}}$$ **Symbols:** **DirichletCharacter** — Dirichlet character; **DirichletL** — Dirichlet L-function. Used by the Compute Engine for simplification. [`c9d117` · Fungrim entry ↗](https://fungrim.org/entry/c9d117) --- $$\mathrm{DirichletL}(1, \mathrm{DirichletCharacter}(3, 2))=\frac{\pi}{\sqrt{27}}$$ **Symbols:** **DirichletCharacter** — Dirichlet character; **DirichletL** — Dirichlet L-function. Used by the Compute Engine for simplification. [`d83109` · Fungrim entry ↗](https://fungrim.org/entry/d83109) --- $$\mathrm{DirichletCharacter}(q, 1, n)=\begin{cases}1&\gcd(n, q)=1\\0&\top\end{cases}$$ **Holds when** $q\in\N^*\land n\in\Z$. **Symbols:** **DirichletCharacter** — Dirichlet character. Used by the Compute Engine for simplification. [`d8c6d1` · Fungrim entry ↗](https://fungrim.org/entry/d8c6d1) --- $$\mathrm{DirichletCharacter}(q, \ell)\lhd n=\mathrm{DirichletCharacter}(q, \ell, n)$$ **Holds when** $q\in\N^*\land\ell\in1..\max(q, 2)-1\land\gcd(\ell, q)=1\land n\in\Z$. **Symbols:** **DirichletCharacter** — Dirichlet character. Used by the Compute Engine for simplification. [`d9a187` · Fungrim entry ↗](https://fungrim.org/entry/d9a187) --- $$\mathrm{PrimitiveDirichletCharacters}(q)=\mathrm{Filter}(\mathrm{DirichletGroup}(q), \chi\mapsto\forall d\in1..q-1, (d\mid q)\implies(\exists a\in0..q-1, \mathrm{CongruentMod}(a, 1, d)\land\gcd(a, q)=1\land\chi(a)\ne1))$$ **Holds when** $q\in\N^*$. **Symbols:** **DirichletGroup** — Dirichlet characters with given modulus; **PrimitiveDirichletCharacters** — Primitive Dirichlet characters with given modulus. Used by the Compute Engine for simplification. **Reference:** T. Apostol (1976), Introduction to Analytic Number Theory, Springer. Chapter 8.7. [`ed65c8` · Fungrim entry ↗](https://fungrim.org/entry/ed65c8) --- $$\mathrm{DirichletCharacter}(4, 3, n)=\begin{cases}1&\mathrm{CongruentMod}(n, 1, 4)\\-1&\mathrm{CongruentMod}(n, 3, 4)\\0&\top\end{cases}$$ **Holds when** $n\in\Z$. **Symbols:** **DirichletCharacter** — Dirichlet character. Used by the Compute Engine for simplification. [`fc4f6a` · Fungrim entry ↗](https://fungrim.org/entry/fc4f6a) --- $$\mathrm{DirichletL}(s, \mathrm{DirichletCharacter}(2^{n}, 1))=(1-2^{-s})\Zeta(s)$$ **Holds when** $n\in\N^*\land s\in\C$. **Symbols:** **DirichletCharacter** — Dirichlet character; **DirichletL** — Dirichlet L-function. Used by the Compute Engine for expansion. [`ff8254` · Fungrim entry ↗](https://fungrim.org/entry/ff8254) --- ## Hurwitz zeta function $$\mathrm{HurwitzZeta}(0, 0)=\frac{1}{2}$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`150b3e` · Fungrim entry ↗](https://fungrim.org/entry/150b3e) --- $$\mathrm{HurwitzZeta}(4, 1)=\frac{\pi^4}{90}$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`2d4828` · Fungrim entry ↗](https://fungrim.org/entry/2d4828) --- $$\mathrm{HurwitzZeta}(3, \frac{1}{6})=91\Zeta(3)+2\sqrt{3}\pi^3$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`2fabeb` · Fungrim entry ↗](https://fungrim.org/entry/2fabeb) --- $$\mathrm{HurwitzZeta}(4, 2)=\frac{\pi^4}{90}-1$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`33690e` · Fungrim entry ↗](https://fungrim.org/entry/33690e) --- $$s\mapsto\mathrm{HurwitzZeta}(s, a)^{\prime}(s)=\mathrm{HurwitzZeta}(s, a, 1)$$ **Holds when** $s\in\C\land s\ne1\land a\in\C\land\Re(a)\gt0$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`3ba544` · Fungrim entry ↗](https://fungrim.org/entry/3ba544) --- $$\mathrm{HurwitzZeta}(0, \frac{1}{2})=0$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`3db90c` · Fungrim entry ↗](https://fungrim.org/entry/3db90c) --- $$\mathrm{HurwitzZeta}(2, \frac{1}{4})=\pi^2+8G$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`3e82c3` · Fungrim entry ↗](https://fungrim.org/entry/3e82c3) --- $$\mathrm{HurwitzZeta}(4, \frac{1}{2})=\frac{\pi^4}{6}$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`4064f5` · Fungrim entry ↗](https://fungrim.org/entry/4064f5) --- $$a\mapsto\mathrm{HurwitzZeta}(s, a)^{\prime}(a)=\mathrm{RisingFactorial}(1-s-r, r)\mathrm{HurwitzZeta}(s+r, a)$$ **Holds when** $s\in\C\land s\ne1\land s+r\ne1\land a\in\C\land\Re(a)\gt0\land r\in\N$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function; **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`40c3e2` · Fungrim entry ↗](https://fungrim.org/entry/40c3e2) --- $$\mathrm{BernoulliPolynomial}(n, z)=-(n\mathrm{HurwitzZeta}(1-n, z))$$ **Holds when** $n\in\N^*\land z\in\C$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial; **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`4228cd` · Fungrim entry ↗](https://fungrim.org/entry/4228cd) --- $$\mathrm{HurwitzZeta}(s, \frac{1}{6})+\mathrm{HurwitzZeta}(s, \frac{5}{6})=(2^{s}-1)(3^{s}-1)\Zeta(s)$$ **Holds when** $s\in\C\land s\ne1$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for expansion. [`4d1f6b` · Fungrim entry ↗](https://fungrim.org/entry/4d1f6b) --- $$\mathrm{HurwitzZeta}(3, 2)=\Zeta(3)-1$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`4dd87c` · Fungrim entry ↗](https://fungrim.org/entry/4dd87c) --- $$\operatorname{Li}_{s}(z)=\frac{(\mathrm{HurwitzZeta}(1-s, \frac{1}{\pi}((-1/2\imaginaryI)\ln(-z))+\frac{1}{2})\imaginaryI^{1-s}+\mathrm{HurwitzZeta}(1-s, \frac{1}{\pi}((1/2\imaginaryI)\ln(-z))+\frac{1}{2})\imaginaryI^{s-1})\Gamma(1-s)}{(2\pi)^{1-s}}$$ **Holds when** $s\in\C\land z\in\C\land z\notin\lbrace0, 1\rbrace\land s\notin\N$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`52ea5f` · Fungrim entry ↗](https://fungrim.org/entry/52ea5f) --- $$\Gamma(z)=\sqrt{2\pi}\exp(\mathrm{HurwitzZeta}(0, z, 1))$$ **Holds when** $z\in\C\setminus\Z_{\le0}$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`53026a` · Fungrim entry ↗](https://fungrim.org/entry/53026a) --- $$\mathrm{HurwitzZeta}(1, a)=\tilde\infty$$ **Holds when** $a\in\C\setminus\Z_{\le0}$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`532f31` · Fungrim entry ↗](https://fungrim.org/entry/532f31) --- $$\mathrm{HurwitzZeta}(2, 1)=\frac{\pi^2}{6}$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`575b8f` · Fungrim entry ↗](https://fungrim.org/entry/575b8f) --- $$\mathrm{HurwitzZeta}(-n, a)=-(\frac{\mathrm{BernoulliPolynomial}(n+1, a)}{n+1})$$ **Holds when** $n\in\N\land a\in\C$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial; **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`5bdba2` · Fungrim entry ↗](https://fungrim.org/entry/5bdba2) --- $$\mathrm{HurwitzZeta}(2, a)=\mathrm{Hypergeometric3F_2}(1, a, a, a+1, a+1, 1)/a^2$$ **Holds when** $a\in\C\setminus\Z_{\le0}$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`6419ac` · Fungrim entry ↗](https://fungrim.org/entry/6419ac) --- $$\mathrm{HurwitzZeta}(s, \frac{1}{2}+n)=(2^{s}-1)\Zeta(s)-2^{s}(\sum_{k=0}^{n-1}((2k+1)^{s})^{-1})$$ **Holds when** $s\in\C\land n\in\N$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`6c3523` · Fungrim entry ↗](https://fungrim.org/entry/6c3523) --- $$\mathrm{HurwitzZeta}(s, n)=\Zeta(s)-(\sum_{k=1}^{n-1}\frac{1}{k^{s}})$$ **Holds when** $s\in\C\land n\in\N^*$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`6e69fc` · Fungrim entry ↗](https://fungrim.org/entry/6e69fc) --- $$\mathrm{HurwitzZeta}(-n, 0)=-(\frac{\mathrm{BernoulliB}(n+1)}{n+1})$$ **Holds when** $n\in\N$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`7dab87` · Fungrim entry ↗](https://fungrim.org/entry/7dab87) --- $$a\mapsto\mathrm{HurwitzZeta}(s, a)^{\prime}(a)=-(s\mathrm{HurwitzZeta}(s+1, a))$$ **Holds when** $s\in\C\land s\notin\lbrace0, 1\rbrace\land a\in\C\land\Re(a)\gt0$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`83065e` · Fungrim entry ↗](https://fungrim.org/entry/83065e) --- $$\mathrm{HurwitzZeta}(n, a)=\frac{\mathrm{PolyGamma}(n-1, a)\times(-1)^{n}}{(n-1)!}$$ **Holds when** $n\in2..\infty\land a\in\C$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`84196a` · Fungrim entry ↗](https://fungrim.org/entry/84196a) --- $$\mathrm{HurwitzZeta}(2, \frac{1}{2})=\frac{\pi^2}{2}$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`868061` · Fungrim entry ↗](https://fungrim.org/entry/868061) --- $$\mathrm{HurwitzZeta}(s, \frac{1}{4})+\mathrm{HurwitzZeta}(s, \frac{3}{4})=2^{s}(2^{s}-1)\Zeta(s)$$ **Holds when** $s\in\C\land s\ne1$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`8bbb6f` · Fungrim entry ↗](https://fungrim.org/entry/8bbb6f) --- $$\mathrm{HurwitzZeta}(3, \frac{1}{2})=7\Zeta(3)$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`9417f4` · Fungrim entry ↗](https://fungrim.org/entry/9417f4) --- $$\mathrm{HurwitzZeta}(2, \frac{3}{4})=\pi^2-8G$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`951f86` · Fungrim entry ↗](https://fungrim.org/entry/951f86) --- $$\mathrm{HurwitzZeta}(2, 2)=\frac{\pi^2}{6}-1$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`ac8d3c` · Fungrim entry ↗](https://fungrim.org/entry/ac8d3c) --- $$\mathrm{HurwitzZeta}(s, 1)=\Zeta(s)$$ **Holds when** $s\in\C$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for expansion. [`af23f7` · Fungrim entry ↗](https://fungrim.org/entry/af23f7) --- $$\mathrm{HurwitzZeta}(s, \frac{1}{2})=(2^{s}-1)\Zeta(s)$$ **Holds when** $s\in\C$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`af7d3d` · Fungrim entry ↗](https://fungrim.org/entry/af7d3d) --- $$\mathrm{HurwitzZeta}(3, \frac{3}{4})=28\Zeta(3)-\pi^3$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`b347d3` · Fungrim entry ↗](https://fungrim.org/entry/b347d3) --- $$\mathrm{HurwitzZeta}(3, 1)=\Zeta(3)$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`b4ed44` · Fungrim entry ↗](https://fungrim.org/entry/b4ed44) --- $$\mathrm{HurwitzZeta}(s, 2)=\Zeta(s)-1$$ **Holds when** $s\in\C$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`b721b4` · Fungrim entry ↗](https://fungrim.org/entry/b721b4) --- $$\mathrm{HurwitzZeta}(s, \frac{3}{2})=(2^{s}-1)\Zeta(s)-2^{s}$$ **Holds when** $s\in\C$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`c6d6e2` · Fungrim entry ↗](https://fungrim.org/entry/c6d6e2) --- $$s\mapsto\mathrm{HurwitzZeta}(s, a)^{\prime}(s)=\mathrm{HurwitzZeta}(s, a, r)$$ **Holds when** $s\in\C\land s\ne1\land a\in\C\land\Re(a)\gt0\land r\in\N$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`d0d03b` · Fungrim entry ↗](https://fungrim.org/entry/d0d03b) --- $$\mathrm{HurwitzZeta}(0, a)=\frac{1}{2}-a$$ **Holds when** $a\in\C$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`d99808` · Fungrim entry ↗](https://fungrim.org/entry/d99808) --- $$\mathrm{HurwitzZeta}(s, a)=\frac{\mathrm{HurwitzZeta}(s, \frac{a}{2})+\mathrm{HurwitzZeta}(s, \frac{a+1}{2})}{2^{s}}$$ **Holds when** $s\in\C\land a\in\C\land s\ne1\land\Re(a)\gt0$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`ebc49c` · Fungrim entry ↗](https://fungrim.org/entry/ebc49c) --- $$\mathrm{HurwitzZeta}(3, \frac{1}{4})=28\Zeta(3)+\pi^3$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`eda0f3` · Fungrim entry ↗](https://fungrim.org/entry/eda0f3) --- $$\mathrm{HurwitzZeta}(3, \frac{5}{6})=91\Zeta(3)-2\sqrt{3}\pi^3$$ **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`edad97` · Fungrim entry ↗](https://fungrim.org/entry/edad97) --- $$\mathrm{GammaLn}(z)=\mathrm{HurwitzZeta}(0, z, 1)+\frac{\ln(2\pi)}{2}$$ **Holds when** $z\in\C\setminus\Z_{\le0}$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`f3b870` · Fungrim entry ↗](https://fungrim.org/entry/f3b870) --- $$\mathrm{HurwitzZeta}(s, 3)=\Zeta(s)-1-\frac{1}{2^{s}}$$ **Holds when** $s\in\C$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`fc6fe0` · Fungrim entry ↗](https://fungrim.org/entry/fc6fe0) --- ## Multiple zeta values $$\mathrm{MultiZetaValue}(3, 3)=\frac{1}{2}(\Zeta(3)^2-\Zeta(6))$$ **Symbols:** **MultiZetaValue** — Multiple zeta value (MZV). Used by the Compute Engine for simplification. [`3a5167` · Fungrim entry ↗](https://fungrim.org/entry/3a5167) --- $$\mathrm{MultiZetaValue}(2, 2)=\frac{3\Zeta(4)}{4}$$ **Symbols:** **MultiZetaValue** — Multiple zeta value (MZV). Used by the Compute Engine for simplification. [`62de01` · Fungrim entry ↗](https://fungrim.org/entry/62de01) --- $$\mathrm{MultiZetaValue}(2, 3)=\frac{9\Zeta(5)}{2}-2\Zeta(2)\Zeta(3)$$ **Symbols:** **MultiZetaValue** — Multiple zeta value (MZV). Used by the Compute Engine for simplification. [`856317` · Fungrim entry ↗](https://fungrim.org/entry/856317) --- $$\mathrm{MultiZetaValue}(3, 2)=3\Zeta(2)\Zeta(3)-\frac{11\Zeta(5)}{2}$$ **Symbols:** **MultiZetaValue** — Multiple zeta value (MZV). Used by the Compute Engine for simplification. [`a5e52e` · Fungrim entry ↗](https://fungrim.org/entry/a5e52e) --- $$\mathrm{MultiZetaValue}(a, b)+\mathrm{MultiZetaValue}(b, a)=\Zeta(a)\Zeta(b)-\Zeta(a+b)$$ **Holds when** $a\in2..\infty\land b\in2..\infty$. **Symbols:** **MultiZetaValue** — Multiple zeta value (MZV). Used by the Compute Engine for simplification. [`da71d3` · Fungrim entry ↗](https://fungrim.org/entry/da71d3) --- $$\mathrm{MultiZetaValue}(4, 2)=\Zeta(3)^2-\frac{4\Zeta(6)}{3}$$ **Symbols:** **MultiZetaValue** — Multiple zeta value (MZV). Used by the Compute Engine for simplification. [`ef2c71` · Fungrim entry ↗](https://fungrim.org/entry/ef2c71) --- $$\mathrm{MultiZetaValue}(s, s)=\frac{1}{2}(\Zeta(s)^2-\Zeta(2s))$$ **Holds when** $s\in2..\infty$. **Symbols:** **MultiZetaValue** — Multiple zeta value (MZV). Used by the Compute Engine for simplification. [`ef8b17` · Fungrim entry ↗](https://fungrim.org/entry/ef8b17) --- ## Riemann zeta function $$\mathrm{KeiperLiLambda}(0)=0$$ **Symbols:** **KeiperLiLambda** — Keiper-Li coefficient. Used by the Compute Engine for simplification. [`081205` · Fungrim entry ↗](https://fungrim.org/entry/081205) --- $$\mathrm{StieltjesGamma}(n, 1)=\mathrm{StieltjesGamma}(n)$$ **Holds when** $n\in\N$. **Symbols:** **StieltjesGamma** — Stieltjes constant. Used by the Compute Engine for expansion. [`51206a` · Fungrim entry ↗](https://fungrim.org/entry/51206a) --- $$\Zeta(-n)=\frac{(-1)^{n}\mathrm{BernoulliB}(n+1)}{n+1}$$ **Holds when** $n\in\Z\land n\ge0$. Used by the Compute Engine for simplification. [`51fd98` · Fungrim entry ↗](https://fungrim.org/entry/51fd98) --- $$\mathrm{RiemannZetaZero}(-n)=\mathrm{RiemannZetaZero}(n)^\star$$ **Holds when** $n\in\Z\land n\ne0$. **Symbols:** **RiemannZetaZero** — Nontrivial zero of the Riemann zeta function. Used by the Compute Engine for simplification. [`60c2ec` · Fungrim entry ↗](https://fungrim.org/entry/60c2ec) --- $$\mathrm{HurwitzZeta}(s, a)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{1}{n!}(\mathrm{StieltjesGamma}(n, a)\times(-1)^{n}(s-1)^{n})$$ **Holds when** $s\in\C\land a\in\C\land a\notin\Z_{\le0}$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function; **StieltjesGamma** — Stieltjes constant. Used by the Compute Engine for simplification. [`60c6da` · Fungrim entry ↗](https://fungrim.org/entry/60c6da) --- $$\mathrm{StieltjesGamma}(n, a+1)=\mathrm{StieltjesGamma}(n, a)-\frac{\ln(a)^{n}}{a}$$ **Holds when** $n\in\N\land a\in\C\land a\notin\Z_{\le0}$. **Symbols:** **StieltjesGamma** — Stieltjes constant. Used by the Compute Engine for simplification. [`687b4d` · Fungrim entry ↗](https://fungrim.org/entry/687b4d) --- $$\Zeta(s^\star)=\Zeta(s)^\star$$ **Holds when** $s\in\C\land s\ne1$. Used by the Compute Engine for expansion. [`69348a` · Fungrim entry ↗](https://fungrim.org/entry/69348a) --- $$\mathrm{StieltjesGamma}(1, \frac{1}{2})=\mathrm{StieltjesGamma}(1)-2\gamma\ln(2)-\ln(2)^2$$ **Symbols:** **StieltjesGamma** — Stieltjes constant. Used by the Compute Engine for simplification. [`70a705` · Fungrim entry ↗](https://fungrim.org/entry/70a705) --- $$\Zeta(2n)=\frac{(-1)^{n+1}\mathrm{BernoulliB}(2n)(2\pi)^{2n}}{2(2n)!}$$ **Holds when** $n\in\Z\land n\ge1$. Used by the Compute Engine for simplification. [`72ccda` · Fungrim entry ↗](https://fungrim.org/entry/72ccda) --- $$\Zeta(s)=(\sum_{k=1}^{N_{var}-1}\frac{1}{k^{s}}+\frac{N_{var}^{1-s}}{s-1}+\frac{\sum_{k=1}^{M}(\mathrm{RisingFactorial}(s, 2k-1)\mathrm{BernoulliB}(2k))/((2k)!N_{var}^{2k-1})+\frac{1}{2}}{N_{var}^{s}})-\int_{N_{var}}^{\infty}\!\frac{\mathrm{RisingFactorial}(s, 2M)\mathrm{BernoulliPolynomial}(2M, t-\lfloor t\rfloor)}{(2M)!t^{2M+s}}\, \mathrm{d}t$$ **Holds when** $s\in\C\land s\ne1\land N_{var}\in\Z\land M\in\Z\land\Re((s+2M)-1)\gt0\land N_{var}\ge1\land M\ge1$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial; **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. **References:** - F. Johansson (2015), Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, Numerical Algorithms 69:253, DOI: 10.1007/s11075-014-9893-1 - F. W. J. Olver, Asymptotics and Special Functions, AK Peters, 1997. Chapter 8. [`792f7b` · Fungrim entry ↗](https://fungrim.org/entry/792f7b) --- $$\mathrm{StieltjesGamma}(0, 1)=\mathrm{StieltjesGamma}(0)=\gamma$$ **Symbols:** **StieltjesGamma** — Stieltjes constant. Used by the Compute Engine for simplification. [`8ae153` · Fungrim entry ↗](https://fungrim.org/entry/8ae153) --- $$\Zeta(2)=\frac{\pi^2}{6}$$ Used by the Compute Engine for simplification. [`a01b6e` · Fungrim entry ↗](https://fungrim.org/entry/a01b6e) --- $$\Zeta(s)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{1}{n!}(\mathrm{StieltjesGamma}(n)\times(-1)^{n}(s-1)^{n})$$ **Holds when** $s\in\C$. **Symbols:** **StieltjesGamma** — Stieltjes constant. Used by the Compute Engine for simplification. [`b1a2e1` · Fungrim entry ↗](https://fungrim.org/entry/b1a2e1) --- $$\mathrm{StieltjesGamma}(0, a)=-\mathrm{Digamma}(a)$$ **Holds when** $a\in\C\land a\notin\Z_{\le0}$. **Symbols:** **StieltjesGamma** — Stieltjes constant. Used by the Compute Engine for simplification. [`b6808d` · Fungrim entry ↗](https://fungrim.org/entry/b6808d) --- $$\mathrm{KeiperLiLambda}(1)=(\frac{\gamma}{2}+1)-\frac{\ln(4\pi)}{2}$$ **Symbols:** **KeiperLiLambda** — Keiper-Li coefficient. Used by the Compute Engine for simplification. [`d8d820` · Fungrim entry ↗](https://fungrim.org/entry/d8d820) --- $$\mathrm{KeiperLiLambda}(n)=\frac{1}{n!}(s\mapsto\ln(2\mathrm{RiemannXi}(s/(s-1)))^{\prime}(0))$$ **Holds when** $n\in\N$. **Symbols:** **KeiperLiLambda** — Keiper-Li coefficient. Used by the Compute Engine for simplification. [`fcab61` · Fungrim entry ↗](https://fungrim.org/entry/fcab61) --- $$\Zeta(-2n)=0$$ **Holds when** $n\in\N^*$. Used by the Compute Engine for simplification. `zeta-trivial-zeros` · _curated identity (not in the upstream Fungrim corpus)_ ---