Fungrim home page

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

d5a598
Symbol: DirichletL L ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function

L-series

604c7c
L ⁣(s,χ)=n=1χ ⁣(n)nsL\!\left(s, \chi\right) = \sum_{n=1}^{\infty} \frac{\chi\!\left(n\right)}{{n}^{s}}
291569
1L ⁣(s,χ)=n=1μ ⁣(n)χ ⁣(n)ns\frac{1}{L\!\left(s, \chi\right)} = \sum_{n=1}^{\infty} \frac{\mu\!\left(n\right) \chi\!\left(n\right)}{{n}^{s}}

Euler product

d088ea
L ⁣(s,χ)=p11χ ⁣(p)psL\!\left(s, \chi\right) = \prod_{p} \frac{1}{1 - \chi\!\left(p\right) {p}^{-s}}
0f96c3
1L ⁣(s,χ)=p(1χ ⁣(p)ps)\frac{1}{L\!\left(s, \chi\right)} = \prod_{p} \left(1 - \frac{\chi\!\left(p\right)}{{p}^{s}}\right)

Hurwitz zeta representation

04217b
Symbol: HurwitzZeta ζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
c31c10
L ⁣(s,χ)=1qsk=1qχ ⁣(k)ζ ⁣(s,kq)L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi\!\left(k\right) \zeta\!\left(s, \frac{k}{q}\right)
4c3678
ζ ⁣(s,kq)=qsφ ⁣(q)χGqχ ⁣(k)L ⁣(s,χ)\zeta\!\left(s, \frac{k}{q}\right) = \frac{{q}^{s}}{\varphi\!\left(q\right)} \sum_{\chi \in G_{q}} \overline{\chi\!\left(k\right)} L\!\left(s, \chi\right)

Principal and non-primitive characters

a9337b
L ⁣(s,χ1(1,))=ζ ⁣(s)L\!\left(s, \chi_{1}(1, \cdot)\right) = \zeta\!\left(s\right)
ff8254
L ⁣(s,χ2n(1,))=(12s)ζ ⁣(s)L\!\left(s, \chi_{{2}^{n}}(1, \cdot)\right) = \left(1 - {2}^{-s}\right) \zeta\!\left(s\right)
629f70
L ⁣(s,χq(1,))=ζ ⁣(s)pq(11ps)L\!\left(s, \chi_{q}(1, \cdot)\right) = \zeta\!\left(s\right) \prod_{p \mid q} \left(1 - \frac{1}{{p}^{s}}\right)
1bd945
L ⁣(s,χ)=L ⁣(s,χ0)pq(1χ0 ⁣(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, \cdot),\,\chi = {\chi}_{0} {\chi}_{1}

Value at 1

6c3fff
L ⁣(1,χ)={~,χ=χq(1,)lims1L ⁣(s,χ),otherwiseL\!\left(1, \chi\right) = \begin{cases} {\tilde \infty}, & \chi = \chi_{q}(1, \cdot)\\\lim_{s \to 1} L\!\left(s, \chi\right), & \text{otherwise}\\ \end{cases}
3d5327
L ⁣(1,χ)0L\!\left(1, \chi\right) \ne 0
23256b
lims1(s1)L ⁣(1,χq(1,))=φ ⁣(q)q\lim_{s \to 1} \left(s - 1\right) L\!\left(1, \chi_{q}(1, \cdot)\right) = \frac{\varphi\!\left(q\right)}{q}
d10029
Symbol: StieltjesGamma γn ⁣(a)\gamma_{n}\!\left(a\right) Stieltjes constant
c2750a
L ⁣(1,χ)=1qk=1q1χ ⁣(k)γ0 ⁣(kq)L\!\left(1, \chi\right) = \frac{1}{q} \sum_{k=1}^{q - 1} \chi\!\left(k\right) \gamma_{0}\!\left(\frac{k}{q}\right)
d83109
L ⁣(1,χ3(2,))=π27L\!\left(1, \chi_{3}(2, \cdot)\right) = \frac{\pi}{\sqrt{27}}
3b8c97
L ⁣(1,χ4(3,))=π4L\!\left(1, \chi_{4}(3, \cdot)\right) = \frac{\pi}{4}
c9d117
L ⁣(1,χ5(4,))=2log ⁣(φ)5L\!\left(1, \chi_{5}(4, \cdot)\right) = \frac{2 \log\!\left(\varphi\right)}{\sqrt{5}}

Value at 0

a07d28
L ⁣(0,χq(1,))={1/2,q=10,otherwiseL\!\left(0, \chi_{q}(1, \cdot)\right) = \begin{cases} -1 / 2, & q = 1\\0, & \text{otherwise}\\ \end{cases}
789ca4
L ⁣(0,χ)=1qk=1qkχ ⁣(k)L\!\left(0, \chi\right) = -\frac{1}{q} \sum_{k=1}^{q} k \chi\!\left(k\right)
fad52f
(χ ⁣(1)=1)    (L ⁣(0,χ)=0)\left(\chi\!\left(-1\right) = 1\right) \implies \left(L\!\left(0, \chi\right) = 0\right)

Values at negative integers

cb5d51
Symbol: GeneralizedBernoulliB Bn,χB_{n,\chi} Generalized Bernoulli number
e44796
Bn,χ=a=1qχ ⁣(a)k=0n(nk)Bkankqk1B_{n,\chi} = \sum_{a=1}^{q} \chi\!\left(a\right) \sum_{k=0}^{n} {n \choose k} B_{k} {a}^{n - k} {q}^{k - 1}
3e0817
Bn,χ=qn1a=1qχ ⁣(a)Bn ⁣(aq)B_{n,\chi} = {q}^{n - 1} \sum_{a=1}^{q} \chi\!\left(a\right) B_{n}\!\left(\frac{a}{q}\right)
f7a866
B0,χ={φ(q)/q,χ=χq(1,)0,otherwiseB_{0,\chi} = \begin{cases} \varphi\left(q\right) / q, & \chi = \chi_{q}(1, \cdot)\\0, & \text{otherwise}\\ \end{cases}
d69b41
a=1qχ ⁣(a)zeazeqz1=n=0Bn,χznn!\sum_{a=1}^{q} \chi\!\left(a\right) \frac{z {e}^{a z}}{{e}^{q z} - 1} = \sum_{n=0}^{\infty} B_{n,\chi} \frac{{z}^{n}}{n !}
f5c3c5
L ⁣(n,χ)=Bn+1,χn+1L\!\left(-n, \chi\right) = -\frac{B_{n + 1,\chi}}{n + 1}

Zeros

Nontrivial zeros

3f96c1
Symbol: DirichletLZero ρn,χ\rho_{n, \chi} Nontrivial zero of Dirichlet L-function
dc593e
Symbol: GeneralizedRiemannHypothesis GRH\operatorname{GRH} Generalized Riemann hypothesis
982e3b
0<Re ⁣(ρn,χ)<10 \lt \operatorname{Re}\!\left(\rho_{n, \chi}\right) \lt 1
2a34c3
zerossC,0<Re(s)<1L ⁣(s,χ)={ρn,χ:nZ{0}}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 \lt \operatorname{Re}\left(s\right) \lt 1} L\!\left(s, \chi\right) = \left\{ \rho_{n, \chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}
214a91
Re ⁣(ρn,χ)=12\operatorname{Re}\!\left(\rho_{n, \chi}\right) = \frac{1}{2}

Trivial zeros

bc755b
zerossCL ⁣(s,χ)=(zerossC,Re(s)0L ⁣(s,χ)){ρn,χ:nZ{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}\left(s\right) \le 0} L\!\left(s, \chi\right)\right) \cup \left\{ \rho_{n, \chi} : n \in \mathbb{Z} \setminus \left\{0\right\} \right\}
9ba78a
zerossC,Re(s)0L ⁣(s,χ)={{2n:nZ1},q=1{2n:nZ0},χ(1)=1,q1{2n1:nZ0},χ(1)=1\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,\operatorname{Re}\left(s\right) \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\left(-1\right) = 1,\,q \ne 1\\\left\{ -2 n - 1 : n \in \mathbb{Z}_{\ge 0} \right\}, & \chi\left(-1\right) = -1\\ \end{cases}

Conjugate symmetry

7c86d5
L ⁣(s,χ)=L ⁣(s,χ)L\!\left(s, \overline{\chi}\right) = \overline{L\!\left(\overline{s}, \chi\right)}
50adea
L ⁣(s,χ)=L ⁣(s,χ)L\!\left(\overline{s}, \chi\right) = \overline{L\!\left(s, \overline{\chi}\right)}
97fe89
L ⁣(s,χ)=L ⁣(s,χ)L\!\left(\overline{s}, \overline{\chi}\right) = \overline{L\!\left(s, \chi\right)}

Functional equation

cc6a5a
Symbol: DirichletLambda Λ ⁣(s,χ)\Lambda\!\left(s, \chi\right) Completed Dirichlet L-function
b788a1
Λ ⁣(s,χ)=β(qπ)(s+a)/2Γ ⁣(s+a2)L ⁣(s,χ)   where a=1χ ⁣(1)2,β={s(s1),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\!\left(-1\right)}{2},\,\beta = \begin{cases} s \left(s - 1\right), & q = 1\\1, & \text{otherwise}\\ \end{cases}
11a763
Symbol: GaussSum Gq ⁣(χ)G_{q}\!\left(\chi\right) Gauss sum
62f12c
Gq ⁣(χ)=n=1qχ ⁣(n)e2πin/qG_{q}\!\left(\chi\right) = \sum_{n=1}^{q} \chi\!\left(n\right) {e}^{2 \pi i n / q}
b78a50
Gq ⁣(χ)=q\left|G_{q}\!\left(\chi\right)\right| = \sqrt{q}
288207
Λ ⁣(s,χ)=εΛ ⁣(1s,χ)   where a=1χ ⁣(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\!\left(-1\right)}{2},\,\varepsilon = \frac{G_{q}\!\left(\chi\right)}{{i}^{a} \sqrt{q}}

Analytic properties

8533f5
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\}
97f631
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\}
ea8c55
Poles ⁣(L ⁣(s,χ),s,C{~})={{1},χ=χq(1,){},otherwise\operatorname{Poles}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \begin{cases} \left\{1\right\}, & \chi = \chi_{q}(1, \cdot)\\\left\{\right\}, & \text{otherwise}\\ \end{cases}
fe4692
HolomorphicDomain ⁣(L ⁣(s,χ),s,C{~})={C{1},χ=χq(1,)C,otherwise\operatorname{HolomorphicDomain}\!\left(L\!\left(s, \chi\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \begin{cases} \mathbb{C} \setminus \left\{1\right\}, & \chi = \chi_{q}(1, \cdot)\\\mathbb{C}, & \text{otherwise}\\ \end{cases}

Approximations

312147
L ⁣(s,χ)k=1N1χ ⁣(k)ksζ ⁣(Re ⁣(s),N)\left|L\!\left(s, \chi\right) - \sum_{k=1}^{N - 1} \frac{\chi\!\left(k\right)}{{k}^{s}}\right| \le \zeta\!\left(\operatorname{Re}\!\left(s\right), N\right)
4911bd
1L ⁣(s,χ)p<N(1χ ⁣(p)ps)ζ ⁣(Re ⁣(s),N)\left|\frac{1}{L\!\left(s, \chi\right)} - \prod_{p \lt N} \left(1 - \frac{\chi\!\left(p\right)}{{p}^{s}}\right)\right| \le \zeta\!\left(\operatorname{Re}\!\left(s\right), N\right)

Bounds and inequalities

8ff1ff
L ⁣(s,χ)ζ ⁣(Re ⁣(s))\left|L\!\left(s, \chi\right)\right| \le \zeta\!\left(\operatorname{Re}\!\left(s\right)\right)
9b3fde
L ⁣(s,χ)(q1+s2π)(1+ηRe(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}\left(s\right) \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.

2019-07-15 23:42:41.550119 UTC