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Dirichlet L-functions

Table of contents: Definitions - L-series - Euler product - Hurwitz zeta representation - Principal and non-primitive characters - Value at 1 - Value at 0 - Values at negative integers - Zeros - Conjugate symmetry - Functional equation - Analytic properties - Approximations - Bounds and inequalities - Related topics

Definitions

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Symbol: DirichletL — L ⁣(s,χ)L\!\left(s, \chi\right) — Dirichlet L-function

L-series

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L ⁣(s,χ)=∑n=1∞χ(n)nsL\!\left(s, \chi\right) = \sum_{n=1}^{\infty} \frac{\chi(n)}{{n}^{s}}
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1L ⁣(s,χ)=∑n=1∞μ(n)χ(n)ns\frac{1}{L\!\left(s, \chi\right)} = \sum_{n=1}^{\infty} \frac{\mu(n) \chi(n)}{{n}^{s}}

Euler product

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L ⁣(s,χ)=∏p11−χ(p)p−sL\!\left(s, \chi\right) = \prod_{p} \frac{1}{1 - \chi(p) {p}^{-s}}
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1L ⁣(s,χ)=∏p(1−χ(p)ps)\frac{1}{L\!\left(s, \chi\right)} = \prod_{p} \left(1 - \frac{\chi(p)}{{p}^{s}}\right)

Hurwitz zeta representation

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Symbol: HurwitzZeta — ζ ⁣(s,a)\zeta\!\left(s, a\right) — Hurwitz zeta function
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L ⁣(s,χ)=1qs∑k=1qχ(k)ζ ⁣(s,kq)L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi(k) \zeta\!\left(s, \frac{k}{q}\right)
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ζ ⁣(s,kq)=qsφ(q)∑χ∈Gqχ(k)‾L ⁣(s,χ)\zeta\!\left(s, \frac{k}{q}\right) = \frac{{q}^{s}}{\varphi(q)} \sum_{\chi \in G_{q}} \overline{\chi(k)} L\!\left(s, \chi\right)

Principal and non-primitive characters

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L ⁣(s,χ1 . 1)=ζ ⁣(s)L\!\left(s, \chi_{1 \, . \, 1}\right) = \zeta\!\left(s\right)
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L ⁣(s,χ2n . 1)=(1−2−s)ζ ⁣(s)L\!\left(s, \chi_{{2}^{n} \, . \, 1}\right) = \left(1 - {2}^{-s}\right) \zeta\!\left(s\right)
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L ⁣(s,χq . 1)=ζ ⁣(s)∏p∣q(1−1ps)L\!\left(s, \chi_{q \, . \, 1}\right) = \zeta\!\left(s\right) \prod_{p \mid q} \left(1 - \frac{1}{{p}^{s}}\right)
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L ⁣(s,χ)=L ⁣(s,χ0)∏p∣q(1−χ0 ⁣(p)ps)   where χ1=χq . 1,  χ=χ0χ1L\!\left(s, \chi\right) = L\!\left(s, {\chi}_{0}\right) \prod_{p \mid q} \left(1 - \frac{{\chi}_{0}\!\left(p\right)}{{p}^{s}}\right)\; \text{ where } {\chi}_{1} = \chi_{q \, . \, 1},\;\chi = {\chi}_{0} {\chi}_{1}

Value at 1

Related topic: Stieltjes constants
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L ⁣(1,χ)={∞~,χ=χq . 1lim⁡s→1L ⁣(s,χ),otherwiseL\!\left(1, \chi\right) = \begin{cases} {\tilde \infty}, & \chi = \chi_{q \, . \, 1}\\\lim_{s \to 1} L\!\left(s, \chi\right), & \text{otherwise}\\ \end{cases}
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L ⁣(1,χ)≠0L\!\left(1, \chi\right) \ne 0
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L ⁣(1,χ)∉Q‾L\!\left(1, \chi\right) \notin \overline{\mathbb{Q}}
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lim⁡s→1(s−1)L ⁣(1,χq . 1)=φ(q)q\lim_{s \to 1} \left(s - 1\right) L\!\left(1, \chi_{q \, . \, 1}\right) = \frac{\varphi(q)}{q}
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L ⁣(1,χ)=−1q∑k=1q−1χ(k)ψ ⁣(kq)L\!\left(1, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q - 1} \chi(k) \psi\!\left(\frac{k}{q}\right)
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L ⁣(1,χ3 . 2)=π27L\!\left(1, \chi_{3 \, . \, 2}\right) = \frac{\pi}{\sqrt{27}}
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L ⁣(1,χ4 . 3)=π4L\!\left(1, \chi_{4 \, . \, 3}\right) = \frac{\pi}{4}
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L ⁣(1,χ5 . 4)=2log⁡(φ)5L\!\left(1, \chi_{5 \, . \, 4}\right) = \frac{2 \log(\varphi)}{\sqrt{5}}

Value at 0

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L ⁣(0,χq . 1)={−12,q=10,otherwiseL\!\left(0, \chi_{q \, . \, 1}\right) = \begin{cases} -\frac{1}{2}, & q = 1\\0, & \text{otherwise}\\ \end{cases}
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L ⁣(0,χ)=−1q∑k=1qkχ(k)L\!\left(0, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q} k \chi(k)
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(χ(−1)=1)    ⟹    (L ⁣(0,χ)=0)\left(\chi(-1) = 1\right) \;\implies\; \left(L\!\left(0, \chi\right) = 0\right)

Values at negative integers

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Symbol: GeneralizedBernoulliB — Bn,χB_{n,\chi} — Generalized Bernoulli number
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Bn,χ=∑a=1qχ(a)∑k=0n(nk)Bkan−kqk−1B_{n,\chi} = \sum_{a=1}^{q} \chi(a) \sum_{k=0}^{n} {n \choose k} B_{k} {a}^{n - k} {q}^{k - 1}
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Bn,χ=qn−1∑a=1qχ(a)Bn ⁣(aq)B_{n,\chi} = {q}^{n - 1} \sum_{a=1}^{q} \chi(a) B_{n}\!\left(\frac{a}{q}\right)
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B0,χ={φ(q)q,χ=χq . 10,otherwiseB_{0,\chi} = \begin{cases} \frac{\varphi(q)}{q}, & \chi = \chi_{q \, . \, 1}\\0, & \text{otherwise}\\ \end{cases}
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∑a=1qχ(a)zeazeqz−1=∑n=0∞Bn,χznn!\sum_{a=1}^{q} \chi(a) \frac{z {e}^{a z}}{{e}^{q z} - 1} = \sum_{n=0}^{\infty} B_{n,\chi} \frac{{z}^{n}}{n !}
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L ⁣(−n,χ)=−Bn+1,χn+1L\!\left(-n, \chi\right) = -\frac{B_{n + 1,\chi}}{n + 1}

Zeros

Nontrivial zeros

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Symbol: DirichletLZero — ρn,χ\rho_{n,\chi} — Nontrivial zero of Dirichlet L-function
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Symbol: GeneralizedRiemannHypothesis — GRH⁡\operatorname{GRH} — Generalized Riemann hypothesis
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(GRH⁡)  ⟺  (Re⁡ ⁣(ρn,χ)=12   for all (q,χ,n) with q∈Z≥1  and⁡  χ∈Gq  and⁡  n∈Z∖{0})\left(\operatorname{GRH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2} \;\text{ for all } \left(q, \chi, n\right) \text{ with } q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \setminus \left\{0\right\}\right)
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0<Re⁡ ⁣(ρn,χ)<10 < \operatorname{Re}\!\left(\rho_{n,\chi}\right) < 1
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zeros⁡ s∈C, 0<Re⁡(s)<1L ⁣(s,χ)={ρn,χ:n∈Z∖{0}}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 < \operatorname{Re}(s) < 1} L\!\left(s, \chi\right) = \left\{ \rho_{n,\chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}
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Re⁡ ⁣(ρn,χ)=12\operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2}

Trivial zeros

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zeros⁡ s∈CL ⁣(s,χ)=(zeros⁡ s∈C, Re⁡(s)≤0L ⁣(s,χ))∪{ρn,χ:n∈Z∖{0}}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} L\!\left(s, \chi\right) = \left(\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}(s) \le 0} L\!\left(s, \chi\right)\right) \cup \left\{ \rho_{n,\chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}
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zeros⁡ s∈C, Re⁡(s)≤0L ⁣(s,χ)={{−2n:n∈Z≥1},q=1{−2n:n∈Z≥0},χ(−1)=1  and⁡  q≠1{−2n−1:n∈Z≥0},χ(−1)=−1\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}(s) \le 0} L\!\left(s, \chi\right) = \begin{cases} \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}, & q = 1\\\left\{ -2 n : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = 1 \;\mathbin{\operatorname{and}}\; q \ne 1\\\left\{ -2 n - 1 : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi(-1) = -1\\ \end{cases}

Conjugate symmetry

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L ⁣(s,χ‾)=L ⁣(s‾,χ)‾L\!\left(s, \overline{\chi}\right) = \overline{L\!\left(\overline{s}, \chi\right)}
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L ⁣(s‾,χ)=L ⁣(s,χ‾)‾L\!\left(\overline{s}, \chi\right) = \overline{L\!\left(s, \overline{\chi}\right)}
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L ⁣(s‾,χ‾)=L ⁣(s,χ)‾L\!\left(\overline{s}, \overline{\chi}\right) = \overline{L\!\left(s, \chi\right)}

Functional equation

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Symbol: DirichletLambda — Λ ⁣(s,χ)\Lambda\!\left(s, \chi\right) — Completed Dirichlet L-function
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Λ ⁣(s,χ)=β(qπ)(s+a)/2Γ ⁣(s+a2)L ⁣(s,χ)   where a=1−χ(−1)2,  β={s(s−1),q=11,otherwise\Lambda\!\left(s, \chi\right) = \beta {\left(\frac{q}{\pi}\right)}^{\left( s + a \right) / 2} \Gamma\!\left(\frac{s + a}{2}\right) L\!\left(s, \chi\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\beta = \begin{cases} s \left(s - 1\right), & q = 1\\1, & \text{otherwise}\\ \end{cases}
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Symbol: GaussSum — Gq ⁣(χ)G_{q}\!\left(\chi\right) — Gauss sum
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Gq ⁣(χ)=∑n=1qχ(n)e2πin/qG_{q}\!\left(\chi\right) = \sum_{n=1}^{q} \chi(n) {e}^{2 \pi i n / q}
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∣Gq ⁣(χ)∣=q\left|G_{q}\!\left(\chi\right)\right| = \sqrt{q}
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Λ ⁣(s,χ)=εΛ ⁣(1−s,χ‾)   where a=1−χ(−1)2,  ε=Gq ⁣(χ)iaq\Lambda\!\left(s, \chi\right) = \varepsilon \Lambda\!\left(1 - s, \overline{\chi}\right)\; \text{ where } a = \frac{1 - \chi(-1)}{2},\;\varepsilon = \frac{G_{q}\!\left(\chi\right)}{{i}^{a} \sqrt{q}}

Analytic properties

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BranchCuts⁡ ⁣(L ⁣(s,χ),s,C∪{∞~})={}\operatorname{BranchCuts}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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EssentialSingularities⁡ ⁣(L ⁣(s,χ),s,C∪{∞~})={∞~}\operatorname{EssentialSingularities}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}
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poles⁡ s∈C∪{∞~}L ⁣(s,χ)={{1},χ=χq . 1{},otherwise\mathop{\operatorname{poles}\,}\limits_{s \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} L\!\left(s, \chi\right) = \begin{cases} \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\left\{\right\}, & \text{otherwise}\\ \end{cases}
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L ⁣(s,χ) is holomorphic on s∈{C∖{1},χ=χq . 1C,otherwiseL\!\left(s, \chi\right) \text{ is holomorphic on } s \in \begin{cases} \mathbb{C} \setminus \left\{1\right\}, & \chi = \chi_{q \, . \, 1}\\\mathbb{C}, & \text{otherwise}\\ \end{cases}

Approximations

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∣L ⁣(s,χ)−∑k=1N−1χ(k)ks∣≤ζ ⁣(Re⁡(s),N)\left|L\!\left(s, \chi\right) - \sum_{k=1}^{N - 1} \frac{\chi(k)}{{k}^{s}}\right| \le \zeta\!\left(\operatorname{Re}(s), N\right)
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∣1L ⁣(s,χ)−∏p<N(1−χ(p)ps)∣≤ζ ⁣(Re⁡(s),N)\left|\frac{1}{L\!\left(s, \chi\right)} - \prod_{p < N} \left(1 - \frac{\chi(p)}{{p}^{s}}\right)\right| \le \zeta\!\left(\operatorname{Re}(s), N\right)

Bounds and inequalities

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∣L ⁣(s,χ)∣≤ζ ⁣(Re⁡(s))\left|L\!\left(s, \chi\right)\right| \le \zeta\!\left(\operatorname{Re}(s)\right)
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∣L ⁣(s,χ)∣≤(q∣1+s∣2π)(1+η−Re⁡(s))/2ζ ⁣(1+η)\left|L\!\left(s, \chi\right)\right| \le {\left(\frac{q \left|1 + s\right|}{2 \pi}\right)}^{\left( 1 + \eta - \operatorname{Re}(s) \right) / 2} \zeta\!\left(1 + \eta\right)

Related topics: Dirichlet characters, Riemann zeta function, Bernoulli numbers and polynomials

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC