Zeta and L-functions

Part of the Fungrim Identities reference — 79 identities for zeta and l-functions.

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Contents

• Dirichlet characters (15)
• Hurwitz zeta function (40)
• Multiple zeta values (7)
• 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 ↗

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\mathrm{DirichletGroup}(q)=\lbrace\mathrm{DirichletCharacter}(q, \ell), \ell\in1..\max(q, 2)-1\in\gcd(\ell, q)=1\rbrace

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 ↗

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\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 ↗

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\mathrm{ConreyGenerator}(p)=\begin{cases}10&p=40\,487\\7&p=6\,692\,367\,337\\\min(\lbrace a, a\in\N^*\in\mathrm{Count}(\lbrace a^{k}\bmod p, k\in\N\rbrace)=p-1\rbrace)&\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 ↗

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\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 ↗

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\mathrm{ConreyGenerator}(p)=\min(\lbrace a, a\in\N^*\in(\mathrm{Count}(\lbrace a^{k}\bmod p, k\in\N\rbrace)=p-1\land\mathrm{Count}(\lbrace a^{k}\bmod p^2, k\in\N\rbrace)=p(p-1))\rbrace)

Holds when p\in\mathrm{Primes}\land p\ge3. Symbols: ConreyGenerator — Conrey generator. Used by the Compute Engine for simplification. 75231e · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\mathrm{PrimitiveDirichletCharacters}(q)=\lbrace\chi, \chi\in\mathrm{DirichletGroup}(q)\in(\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))\rbrace

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 ↗

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\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 ↗

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\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 ↗

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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 ↗

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\mathrm{HurwitzZeta}(4, 1)=\frac{\pi^4}{90}

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. 2d4828 · Fungrim entry ↗

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\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 ↗

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\mathrm{HurwitzZeta}(4, 2)=\frac{\pi^4}{90}-1

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. 33690e · Fungrim entry ↗

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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 ↗

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\mathrm{HurwitzZeta}(0, \frac{1}{2})=0

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. 3db90c · Fungrim entry ↗

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\mathrm{HurwitzZeta}(2, \frac{1}{4})=\pi^2+8G

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. 3e82c3 · Fungrim entry ↗

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\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 ↗

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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 ↗

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\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 ↗

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\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 ↗

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\mathrm{HurwitzZeta}(3, 2)=\Zeta(3)-1

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. 4dd87c · Fungrim entry ↗

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\mathrm{PolyLog}(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 ↗

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\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 ↗

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\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 ↗

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\mathrm{HurwitzZeta}(2, 1)=\frac{\pi^2}{6}

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. 575b8f · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\mathrm{HurwitzZeta}(-n, 0)=-(\frac{\mathrm{BernoulliB}(n+1)}{n+1})

Holds when n\in\N. Symbols: BernoulliB — Bernoulli number; HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. 7dab87 · Fungrim entry ↗

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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 ↗

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\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 ↗

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\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 ↗

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\mathrm{HurwitzZeta}(3, \frac{1}{2})=7\Zeta(3)

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. 9417f4 · Fungrim entry ↗

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\mathrm{HurwitzZeta}(2, \frac{3}{4})=\pi^2-8G

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. 951f86 · Fungrim entry ↗

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\mathrm{HurwitzZeta}(2, 2)=\frac{\pi^2}{6}-1

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. ac8d3c · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\mathrm{HurwitzZeta}(3, 1)=\Zeta(3)

Symbols: HurwitzZeta — Hurwitz zeta function. Used by the Compute Engine for simplification. b4ed44 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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Riemann zeta function

\mathrm{KeiperLiLambda}(0)=0

Symbols: KeiperLiLambda — Keiper-Li coefficient. Used by the Compute Engine for simplification. 081205 · Fungrim entry ↗

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\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 ↗

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\Zeta(-n)=\frac{(-1)^{n}\mathrm{BernoulliB}(n+1)}{n+1}

Holds when n\in\Z\land n\ge0. Symbols: BernoulliB — Bernoulli number. Used by the Compute Engine for simplification. 51fd98 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\Zeta(s^\star)=\Zeta(s)^\star

Holds when s\in\C\land s\ne1. Used by the Compute Engine for expansion. 69348a · Fungrim entry ↗

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\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 ↗

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\Zeta(2n)=\frac{(-1)^{n+1}\mathrm{BernoulliB}(2n)(2\pi)^{2n}}{2(2n)!}

Holds when n\in\Z\land n\ge1. Symbols: BernoulliB — Bernoulli number. Used by the Compute Engine for simplification. 72ccda · Fungrim entry ↗

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\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: BernoulliB — Bernoulli number; 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 ↗

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\mathrm{StieltjesGamma}(0, 1)=\mathrm{StieltjesGamma}(0)=\gamma

Symbols: StieltjesGamma — Stieltjes constant. Used by the Compute Engine for simplification. 8ae153 · Fungrim entry ↗

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\Zeta(2)=\frac{\pi^2}{6}

Used by the Compute Engine for simplification. a01b6e · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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)

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