L-Functions — Dirichlet and Dedekind Zeta

⚡ Key Takeaways

Dirichlet L-functions $L(s,\chi)=\sum_{n\geq 1}\chi(n)n^{-s}$ and the Dedekind zeta function $\zeta_K(s)=\sum_{\mathfrak{a}}N(\mathfrak{a})^{-s}$ generalize the Riemann zeta function to arithmetic progressions and number fields, respectively. They carry an Euler product, analytic continuation, a functional equation, and encode deep arithmetic — primes in progressions (Dirichlet's theorem) and the class number / regulator (analytic class number formula). Used throughout analytic and algebraic number theory. Foundations: Dirichlet (1837), Dedekind (1870s), Hecke (1917).

💡 The Core Insight

Arithmetic local data $\Rightarrow$ Euler product $\Rightarrow$ global analytic object. A multiplicative coefficient sequence becomes a product over primes; the product converges in a half-plane, continues meromorphically, and satisfies a symmetry $s\leftrightarrow 1-s$ once completed with Gamma factors. The values and zeros of the resulting function then read off arithmetic: distribution of primes, class numbers, units. The unifying principle is that both $L(s,\chi)$ and $\zeta_K(s)$ are special cases of Hecke L-functions, and the factorization $\zeta_K=\prod_\chi L(s,\chi)$ links the two families.

DefinitionDirichlet Character

Let $q\geq 1$ . A Dirichlet character modulo $q$ is a completely multiplicative function $\chi:\mathbb{Z}\to\mathbb{C}$ that is periodic mod $q$ , supported on integers coprime to $q$ , induced from a group homomorphism

1

\quad \chi:(\mathbb{Z}/q\mathbb{Z})^\times \longrightarrow \mathbb{C}^\times,\qquad \chi(n)=0 \text{ if } \gcd(n,q)>1.

Derivation hint: $(\mathbb{Z}/q\mathbb{Z})^\times$ is a finite abelian group of order $\varphi(q)$ ; its characters take values in $\mu_{\varphi(q)}$ , the roots of unity, and complete multiplicativity extends each homomorphism to all of $\mathbb{Z}$ . The principal character $\chi_0$ is $1$ on units and $0$ otherwise. A character is primitive if it is not induced from a character of smaller modulus; the smallest such modulus is the conductor $\mathfrak{f}_\chi$ .

DefinitionDirichlet L-function

For a Dirichlet character $\chi$ mod $q$ and $\operatorname{Re}(s)>1$ ,

2

\quad L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}=\prod_{p}\left(1-\frac{\chi(p)}{p^s}\right)^{-1}.

Derivation hint: complete multiplicativity $\chi(mn)=\chi(m)\chi(n)$ lets each Euler factor be the geometric series $\sum_{k\ge0}\chi(p^k)p^{-ks}$ ; unique factorization of integers reassembles the product into the Dirichlet series. Convergence on $\operatorname{Re}(s)>1$ follows from $|\chi(n)|\le 1$ . When $\chi=\chi_0$ mod $q$ , $L(s,\chi_0)=\zeta(s)\prod_{p\mid q}(1-p^{-s})$ .

DefinitionDedekind Zeta Function

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ . For $\operatorname{Re}(s)>1$ ,

3

\quad \zeta_K(s)=\sum_{\mathfrak{a}\subseteq\mathcal{O}_K,\ \mathfrak{a}\neq 0} \frac{1}{N(\mathfrak{a})^s}=\prod_{\mathfrak{p}}\left(1-\frac{1}{N(\mathfrak{p})^s}\right)^{-1},

where $\mathfrak{a}$ runs over nonzero integral ideals, $\mathfrak{p}$ over nonzero prime ideals, and $N(\mathfrak{a})=[\mathcal{O}_K:\mathfrak{a}]$ is the absolute norm. Derivation hint: $\mathcal{O}_K$ is a Dedekind domain, so every ideal factors uniquely into primes; the Euler product follows exactly as for $\zeta(s)$ , which is the case $K=\mathbb{Q}$ . The norm is multiplicative on ideals, $N(\mathfrak{ab})=N(\mathfrak{a})N(\mathfrak{b})$ .

TheoremAnalytic Continuation and Functional Equation — Dirichlet

Let $\chi$ be primitive mod $q$ , with $\chi(-1)=(-1)^{\mathfrak{a}}$ , $\mathfrak{a}\in\{0,1\}$ . Define the completed L-function

4

\quad \Lambda(s,\chi)=\left(\frac{q}{\pi}\right)^{(s+\mathfrak{a})/2}\Gamma\!\left(\frac{s+\mathfrak{a}}{2}\right)L(s,\chi).

Derivation hint: the Gamma factor is the archimedean Euler factor; $\mathfrak{a}=0$ for even characters (cosine transform), $\mathfrak{a}=1$ for odd (sine transform), matching the parity of the theta series used in the Mellin transform. Then $\Lambda(s,\chi)$ continues to an entire function (for $\chi\neq\chi_0$ ) and satisfies

5

\quad \Lambda(s,\chi)=\frac{\tau(\chi)}{i^{\mathfrak{a}}\sqrt{q}}\,\Lambda(1-s,\overline{\chi}),\qquad \tau(\chi)=\sum_{n=1}^{q}\chi(n)e^{2\pi i n/q}.

Derivation hint: apply the Poisson summation formula to the twisted theta function $\theta_\chi(t)=\sum_n \chi(n)n^{\mathfrak{a}}e^{-\pi n^2 t/q}$ ; the Gauss sum $\tau(\chi)$ arises as the constant of the modular transformation, with $|\tau(\chi)|=\sqrt q$ .

TheoremAnalytic Continuation and Functional Equation — Dedekind

Let $K$ have degree $n=r_1+2r_2$ ( $r_1$ real, $r_2$ complex places), discriminant $d_K$ . Define

6

\quad \Lambda_K(s)=|d_K|^{s/2}\left(\pi^{-s/2}\Gamma\!\left(\tfrac{s}{2}\right)\right)^{r_1}\left(2(2\pi)^{-s}\Gamma(s)\right)^{r_2}\zeta_K(s).

Derivation hint: each real place contributes a $\Gamma(s/2)$ factor (like $\mathbb{R}$ ), each complex place a $\Gamma(s)$ factor (like $\mathbb{C}$ ); the discriminant power normalizes the conductor of $K$ at the archimedean places. Then $\Lambda_K(s)$ extends to a meromorphic function with simple poles only at $s=0,1$ and satisfies

7

\quad \Lambda_K(s)=\Lambda_K(1-s).

Derivation hint: Hecke's proof uses a higher-dimensional theta function over $\mathcal{O}_K$ summed over each ideal class, with Poisson summation on the lattice $\mathcal{O}_K\hookrightarrow K\otimes\mathbb{R}\cong\mathbb{R}^n$ ; symmetry of the theta transformation forces $s\leftrightarrow 1-s$ .

TheoremAnalytic Class Number Formula

$\zeta_K(s)$ has a simple pole at $s=1$ with residue

8

\quad \operatorname{Res}_{s=1}\zeta_K(s)=\frac{2^{r_1}(2\pi)^{r_2}\,h_K\,R_K}{w_K\,\sqrt{|d_K|}},

where $h_K$ is the class number, $R_K$ the regulator, $w_K$ the number of roots of unity in $K$ . Derivation hint: each ideal class contributes a partial zeta function whose residue is computed by lattice-point counting (Minkowski) in the fundamental domain of the unit action; summing over the $h_K$ classes gives the regulator $R_K$ (covolume of the log-unit lattice) and the volume factors $2^{r_1}(2\pi)^{r_2}/\sqrt{|d_K|}$ . Equivalently, at $s=0$ the leading Taylor coefficient is $\zeta_K(s)\sim -\dfrac{h_K R_K}{w_K}s^{\,r_1+r_2-1}$ .

ProofEuler Product and Dirichlet's Theorem on Primes in Progressions

Step 1 — Factorization of $\zeta$ over progressions. For fixed $q$ , the orthogonality relations of characters give, for $\gcd(a,q)=1$ ,

9

\quad \frac{1}{\varphi(q)}\sum_{\chi \bmod q}\overline{\chi}(a)\,\chi(n)=\begin{cases}1 & n\equiv a\ (q)\\ 0 & \text{else.}\end{cases}

Derivation hint: this is column orthogonality in the character table of the abelian group $(\mathbb{Z}/q\mathbb{Z})^\times$ .

Step 2 — Log of the L-function. Taking logarithms of the Euler product [2],

10

\quad \log L(s,\chi)=\sum_p \frac{\chi(p)}{p^s}+O(1)\quad (s\to 1^+).

Derivation hint: expand $-\log(1-\chi(p)p^{-s})$ ; the prime-power terms $k\ge2$ contribute a bounded sum.

Step 3 — Assemble the progression. Combine [9] and [10]:

11

\quad \sum_{p\equiv a\ (q)}\frac{1}{p^s}=\frac{1}{\varphi(q)}\sum_{\chi}\overline{\chi}(a)\log L(s,\chi)+O(1).

Derivation hint: insert the orthogonality projector into the prime sum and swap order of summation.

Step 4 — Pole and non-vanishing. The principal character gives $\log L(s,\chi_0)\sim\log\frac{1}{s-1}\to+\infty$ . For non-principal $\chi$ , $L(s,\chi)$ is finite and nonzero at $s=1$ (the crux), so its log stays bounded. Hence the right side of [11] diverges, forcing infinitely many primes $p\equiv a\ (q)$ . Derivation hint: $L(1,\chi)\neq 0$ for complex $\chi$ follows from the product $\prod_\chi L(s,\chi)=\zeta_K(s)$ having a simple pole, which can absorb at most one zero; the real character case needs a separate argument (Dirichlet's class number formula, or Mertens-type estimate) ruling out a real zero at $s=1$ . $\blacksquare$

∎

ProofFactorization

\zeta_K = \prod_\chi L(s,\chi)

for Abelian

K/\mathbb{Q}

Step 1. Let $K/\mathbb{Q}$ be abelian with conductor $f$ ; by Kronecker–Weber $K\subseteq\mathbb{Q}(\zeta_f)$ . Class field theory gives a surjection $(\mathbb{Z}/f\mathbb{Z})^\times\twoheadrightarrow\operatorname{Gal}(K/\mathbb{Q})$ , dual to a group $X$ of Dirichlet characters.

Step 2 — Compare Euler factors at $p\nmid f$ . A prime $p$ splits in $K$ according to the order of its Frobenius. The local factor of $\zeta_K$ at $p$ is

12

\quad \prod_{\mathfrak{p}\mid p}\left(1-N(\mathfrak{p})^{-s}\right)^{-1}=\prod_{\chi\in X}\left(1-\chi(p)p^{-s}\right)^{-1}.

Derivation hint: both sides equal $\prod_{\mathfrak{p}\mid p}(1-p^{-f_\mathfrak{p}s})^{-1}$ where $f_\mathfrak{p}$ is the residue degree; on the right, $\chi(p)$ ranges over all $g$ -th roots of unity each $f$ times, $fg=[K:\mathbb{Q}]$ , reproducing the same factor by the cyclotomic identity $\prod_\zeta(1-\zeta T)=1-T^f$ .

Step 3 — Global product. Taking products over all $p$ (with the ramified factors matching via primitivity/conductor bookkeeping),

13

\quad \zeta_K(s)=\prod_{\chi\in X}L(s,\chi).

Derivation hint: multiply [12] over all primes; ramified primes are handled by using primitive characters of conductor dividing $f$ so the Euler factors at $p\mid f$ also match.

Consequence. Since $\zeta_K$ has exactly a simple pole at $s=1$ (residue [8]) and $L(s,\chi_0)$ supplies that pole, every non-principal $L(s,\chi)$ in $X$ is finite and nonzero at $s=1$ . This is the cleanest proof that $L(1,\chi)\neq 0$ . $\blacksquare$

∎

PG

Peter Gustav Lejeune Dirichlet

1805–1859

Dirichlet introduced the series [2] in 1837 to prove infinitude of primes in arithmetic progressions, fusing analysis and number theory in a way that founded analytic number theory. He also derived the class number formula for quadratic forms — the first instance of [8] — connecting $L(1,\chi)$ to the count of equivalence classes of binary quadratic forms, anticipating the role of $h_K$ . His pigeonhole principle, Dirichlet unit theorem (giving $R_K$ ), and rigorous Fourier-analytic methods made these L-functions a permanent tool. Dedekind, his intellectual heir, later generalized the zeta function to arbitrary number fields, encoding all of this into $\zeta_K(s)$ .

🧠 Intuition

Think of $\zeta(s)=\prod_p(1-p^{-s})^{-1}$ as the generating bookkeeper of the integers. A Dirichlet character $\chi$ is a "color" assigned to each residue class; $L(s,\chi)$ weighs each prime by its color, so taking a weighted average over all colors selects a single progression — exactly the projection in [9]. The miracle is that the analytic behavior (a pole vs. finite nonzero value at $s=1$ ) translates into a yes/no answer about infinitely many primes. For $\zeta_K(s)$ , replace "integers" by "ideals of $\mathcal{O}_K$ ." A rational prime $p$ may split, ramify, or stay inert in $K$ ; the local Euler factor records which. The pole at $s=1$ measures the density of ideals, and that density is governed by the geometry of the number field: how many units it has ( $R_K$ , $w_K$ ), how its ideal classes spread out ( $h_K$ ), and how "twisted" its embedding into $\mathbb{R}^n$ is ( $\sqrt{|d_K|}$ ). The class number formula [8] is the statement that analysis (residue) = geometry (covolumes).

Figure

Splitting of a rational prime and the Euler factor structure

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ExampleComputing

L(1,\chi)

for the nontrivial character mod 4

Given: $q=4$ , $(\mathbb{Z}/4)^\times=\{1,3\}$ . The nontrivial character $\chi_{-4}$ has $\chi(1)=1,\ \chi(3)=-1,\ \chi(\text{even})=0$ . It is odd ( $\chi(-1)=\chi(3)=-1$ ), conductor $4$ .

Steps: 1. The series is $L(s,\chi_{-4})=1-3^{-s}+5^{-s}-7^{-s}+\cdots$ (the Dirichlet beta function $\beta(s)$ ). 2. At $s=1$ this is the Leibniz series:

14

\quad L(1,\chi_{-4})=1-\tfrac13+\tfrac15-\tfrac17+\cdots=\frac{\pi}{4}.

Derivation hint: integrate $\sum(-1)^k x^{2k}=1/(1+x^2)$ from $0$ to $1$ , giving $\arctan(1)=\pi/4$ ; Abel summation justifies the value at the boundary. 3. Class number check. For $K=\mathbb{Q}(i)$ , $\zeta_K=\zeta(s)L(s,\chi_{-4})$ . The residue at $s=1$ is $\operatorname{Res}\zeta(s)\cdot L(1,\chi_{-4})=1\cdot\frac\pi4=\frac\pi4$ . Formula [8] with $r_1=0,r_2=1,h_K=1,w_K=4,d_K=-4$ gives $\frac{(2\pi)^1\cdot1\cdot1}{4\cdot\sqrt4}=\frac{2\pi}{8}=\frac\pi4.$ ✓

Answer: $\boxed{L(1,\chi_{-4})=\pi/4}$ , consistent with $\mathbb{Q}(i)$ having $h=1$ .

ExampleDedekind zeta of

K=\mathbb{Q}(\sqrt{-5})

and the class number

Given: $K=\mathbb{Q}(\sqrt{-5})$ , ring $\mathcal{O}_K=\mathbb{Z}[\sqrt{-5}]$ , discriminant $d_K=-20$ , so the associated quadratic character is $\chi_{-20}$ mod $20$ . Here $r_1=0,r_2=1,w_K=2$ .

Steps: 1. Factor: $\zeta_K(s)=\zeta(s)\,L(s,\chi_{-20})$ . 2. Residue at $s=1$ : $\operatorname{Res}_{s=1}\zeta_K=L(1,\chi_{-20})$ . 3. Dirichlet's class number formula for an imaginary quadratic field of discriminant $d<-4$ :

15

\quad L(1,\chi_d)=\frac{2\pi h_K}{w_K\sqrt{|d|}}.

Derivation hint: specialize [8] with $r_1=0,r_2=1$ ; equivalently sum $\chi_d(n)/n$ via the finite Hurwitz/Gauss-sum evaluation $L(1,\chi_d)=-\frac{\pi}{|d|^{3/2}}\sum_{a=1}^{|d|}\chi_d(a)\,a$ for odd $\chi$ . 4. With $|d|=20$ , $w_K=2$ , and the known $h_K=2$ :

16

\quad L(1,\chi_{-20})=\frac{2\pi\cdot 2}{2\sqrt{20}}=\frac{2\pi}{\sqrt{20}}=\frac{\pi}{\sqrt5}\approx 1.4050.

Derivation hint: plug numbers into [15]; $\sqrt{20}=2\sqrt5$ cancels one factor of $2$ .

Answer: $\boxed{h_K=2,\ \operatorname{Res}_{s=1}\zeta_K=\pi/\sqrt5\approx1.405}$ . The nontrivial class is $(2,1+\sqrt{-5})$ — this is why $6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})$ has non-unique element factorization.

⚠️ Common mistake

" $L(s,\chi_0)$ is entire." False. The principal character's L-function differs from $\zeta(s)$ only by finitely many Euler factors and therefore inherits the pole at $s=1$ . Only non-principal characters give entire $L(s,\chi)$ .
"The functional equation relates $L(s,\chi)$ to itself." It relates $L(s,\chi)$ to $L(1-s,\overline\chi)$ — the conjugate character — unless $\chi$ is real.
" $\zeta_K(s)/\zeta(s)$ is always entire." This is the Aramata–Brauer theorem and holds for Galois (or general) extensions, but the proof is non-trivial; for non-normal $K$ it relies on Brauer induction, and for general Artin L-functions the analogous holomorphy is the Artin conjecture, still open.
Functional equations require primitive characters. The clean form [5] holds only for primitive $\chi$ ; an imprimitive character's $L$ -function differs from the primitive one by Euler factors at the extra ramified primes and does not satisfy a clean equation.
The Gauss sum $\tau(\chi)$ is not always $\sqrt q$ in sign/phase. Only $|\tau(\chi)|=\sqrt q$ for primitive $\chi$ ; the argument (root number) is generally a subtle algebraic number.

Counterexample

Not every "zeta-like" series has an Euler product or continuation. Consider $\sum_n a_n n^{-s}$ with $a_n=(-1)^{\Omega(n)}$ (Liouville-type but with a twist that breaks complete multiplicativity), or random sign sequences: these may converge in a half-plane but possess no Euler product and need not continue past $\operatorname{Re}(s)=\sigma_c$ . The Euler product — i.e. multiplicativity of the coefficients — is the essential structural input; without it, none of [4]–[8] survive. Concretely, $\sum (-1)^{n} n^{-s}=(2^{1-s}-1)\zeta(s)$ does continue, but only because it secretly equals $\zeta$ times an entire factor; a generic sign sequence has natural boundary $\operatorname{Re}(s)=1$ .

connection

Dirichlet and Dedekind L-functions are the abelian tier of a vast hierarchy. Both are special cases of Hecke L-functions (attached to Hecke Größencharaktere), which themselves are degree-1 cases of Artin L-functions (attached to Galois representations) and conjecturally of Automorphic L-functions via the Langlands program. The factorization $\zeta_K=\prod_\chi L(s,\chi)$ for abelian $K$ is the prototype of the Artin Formalism $\zeta_K(s)=\prod_\rho L(s,\rho)^{\dim\rho}$ over irreducible Galois reps. The non-vanishing $L(1,\chi)\ne0$ is the engine of the Prime Number Theorem for Arithmetic Progressions, while the residue formula [8] is the abelian shadow of Beilinson Conjectures and the Birch and Swinnerton-Dyer Conjecture for elliptic curves. The functional equation is the analytic incarnation of Tate's thesis (Adelic Harmonic Analysis), recasting [4]–[7] as a global Fourier duality on idèles.

🔬 Where this appears

⏳ History

- 1837 — Dirichlet introduces

L(s,\chi)

and proves infinitely many primes in

a+q\mathbb{Z}

for

\gcd(a,q)=1

, founding analytic number theory; he establishes

L(1,\chi)\ne0

via his class number formula for binary quadratic forms.

- 1859 — Riemann treats

\zeta(s)

analytically, supplying the template (continuation, functional equation, zeros) later imitated by Dirichlet and Dedekind L-functions.

- 1871–1879 — Dedekind defines ideals,

\mathcal{O}_K

, and the zeta function

\zeta_K(s)

in his supplements to Dirichlet's lectures, proving its Euler product over prime ideals.

- 1903 — Landau rigorously proves analytic continuation and the class number formula residue for

\zeta_K

, and gives the prime ideal theorem.

- 1917 — Hecke proves the functional equation [7] for

\zeta_K

and introduces general Hecke characters and their L-functions, unifying [4] and [6] via theta functions.

- 1923 — Artin defines Artin L-functions for Galois representations; the factorization

\zeta_K=\prod L(s,\rho)

generalizes [13]; Artin's holomorphy conjecture is posed.

- 1931 — Aramata; 1947 — Brauer prove

\zeta_K/\zeta

is entire for Galois

K/\mathbb{Q}

, partial progress toward Artin's conjecture via Brauer induction.

- 1950 — Tate (thesis, published 1967) recasts the functional equation as adelic Fourier analysis, giving the conceptual proof of [5] and [7].

- 1967– — Langlands subsumes all of these into the conjectural correspondence between Galois/automorphic representations, making Dirichlet and Dedekind L-functions the GL(1) base case.

open-problem

Generalized Riemann Hypothesis (GRH). All nontrivial zeros of $L(s,\chi)$ (and $\zeta_K$ ) are conjectured to lie on $\operatorname{Re}(s)=1/2$ . Open for every nontrivial case; would give optimal error terms in [11]-type estimates.
Siegel zeros. It is not ruled out that a real primitive $\chi$ has a real zero $\beta$ extremely close to $1$ ("Landau–Siegel zero"). Such a zero would make class numbers of imaginary quadratic fields anomalously small and degrade effective bounds (Siegel's theorem is famously ineffective precisely because of this).
Artin's holomorphy conjecture. Whether non-abelian Artin L-functions are entire (away from $s=1$ ) is open; for abelian $K$ it is the trivial factorization [13], but the general case is part of the Langlands program.
Stark's conjectures. The precise algebraic nature of $L'(0,\chi)$ and the associated Stark units remains conjectural beyond rank-1 abelian cases.
Special value transcendence. While $L(1,\chi)$ for odd $\chi$ is an algebraic multiple of $\pi$ , the arithmetic nature of $L(2,\chi)$ , $L(3,\chi)$ (Beilinson regulators, polylogarithms) is largely conjectural.

L-Functions — Dirichlet and Dedekind Zeta

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits