21.03.02 · number-theory / l-functions

Dirichlet $L$ -functions $L (s, χ)$

shipped3 tiersLean: partial

Anchor (Master): Dirichlet 1837 *Abh. Königl. Preuss. Akad.* 45-81 (originator paper: Beweis des Satzes, dass jede unbegrenzte arithmetische Progression…); Dirichlet 1839/40 *J. reine angew. Math.* 19, 21 (continuation, the class-number formula); Davenport *Multiplicative Number Theory* 3rd ed. (Springer GTM 74, 2000); Iwaniec-Kowalski *Analytic Number Theory* (AMS Colloquium Publications 53, 2004) Ch. 4-5; Apostol *Introduction to Analytic Number Theory* (Springer UTM, 1976) Ch. 6-7; Selberg 1949 *Canad. J. Math.* 2, 66-78 (elementary proof of PNT for arithmetic progressions); Tate 1950 Princeton PhD thesis published 1967 in Cassels-Fröhlich *Algebraic Number Theory* (Thompson, Washington) Ch. XV (Tate's thesis: adelic / Fourier-analytic reformulation, functional equations of Hecke $L$); Bump *Automorphic Forms and Representations* (Cambridge Studies in Advanced Mathematics 55, 1997) Ch. 1, 3 (modern reformulation of Dirichlet $L$ as $\mathrm{GL}_1$ automorphic $L$); Manin-Panchishkin *Introduction to Modern Number Theory* (Springer Encyclopaedia of Mathematical Sciences 49, 2nd ed. 2005) Ch. 6 of Part II (the $L$-function spine of arithmetic geometry)

Intuition [Beginner]

Euclid proved that there are infinitely many primes by a one-line argument: multiply all known primes together, add one, and the result has a new prime factor. Two thousand years later one can ask a refined question. The primes other than $2$ are all odd; written in base ten, they end in one of $1, 3, 7, 9$ . Are there infinitely many primes ending in $7$ ? In $3$ ? More generally, fix a starting number $a$ and a step size $m$ with no common factor; the arithmetic progression $a, a + m, a + 2 m, a + 3 m, \dots$ runs through numbers congruent to $a$ modulo $m$ . Are there infinitely many primes in this sequence?

In 1837 Dirichlet proved the answer is yes — for every starting point $a$ coprime to the step $m$ , the progression contains infinitely many primes. Euclid's argument does not extend; Dirichlet's proof is the first time in mathematics that an analytic tool — an infinite sum thought of as a function of a complex variable — settled a question about whole numbers. The tool is a generalisation of the zeta function: Dirichlet attached a numerical "colour" to each integer using a special function $χ$ , and built the corresponding "coloured zeta" $L (s, χ)$ .

The picture is direct. Treat $L (s, χ)$ as a zeta function that tracks not just the integer $n$ but also a label on $n$ — for the simplest example, the label is $+ 1$ if $n$ is one more than a multiple of $4$ , and $- 1$ if $n$ is one less than a multiple of $4$ . The labels are designed so that the labels of products are the products of labels. Dirichlet's miracle is that the behaviour of $L (s, χ)$ at $s = 1$ — whether the function is zero there or non-zero — controls the distribution of primes across the residue classes mod $m$ .

Visual [Beginner]

The picture is the colour-by-residue grid for the small case $m = 4$ . The non-zero residues mod $4$ are $1$ and $3$ ; the two functions $χ$ on these residues are the principal character $χ_{0} (1) = χ_{0} (3) = 1$ and the non-principal character $χ_{1} (1) = + 1, χ_{1} (3) = - 1$ . Each integer coprime to $4$ inherits its colour from its residue: $5, 13, 17, 21$ are coloured $+ 1$ (they are $\equiv 1 mod 4$ ); $3, 7, 11, 15$ are coloured $- 1$ (they are $\equiv 3 mod 4$ ). The $L$ -function $L (s, χ_{1})$ is the sum $1/ 1^{s} - 1/ 3^{s} + 1/ 5^{s} - 1/ 7^{s} + \dots$ — an alternating sum across the odd integers.

$A grid of integers $1$ through $20$ coloured by their residue mod 4: $+1$ residues in one colour, $-1$ residues in another, even numbers greyed out. Below the grid is the corresponding $L$-series $L(s, \chi_1) = 1/1^s - 1/3^s + 1/5^s - \cdots$ as an alternating sum.$

At $s = 1$ this sum equals $π /4$ — a famous identity due to Leibniz. The value is non-zero, and that single non-zero fact is the reason there are infinitely many primes $\equiv 1 mod 4$ and infinitely many primes $\equiv 3 mod 4$ .

Worked example [Beginner]

Count primes in the two residue classes mod $4$ up to $30$ .

Step 1. List the primes up to $30$ : $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$ . Drop $2$ (the only even prime) since it does not interact with the colouring mod $4$ .

Step 2. Sort by residue mod $4$ . Primes congruent to $1 mod 4$ : $5, 13, 17, 29$ (four primes). Primes congruent to $3 mod 4$ : $3, 7, 11, 19, 23$ (five primes).

Step 3. Both classes contain several primes already by $30$ . Dirichlet proved both classes contain infinitely many primes — the count up to $x$ in each class grows like $x / (2 ln x)$ as $x$ goes to infinity, half the total prime count (the prime-number theorem in arithmetic progressions).

What this tells us: the primes spread across the residue classes mod $4$ in equal proportions, and Dirichlet's $L$ -function $L (s, χ_{1}) = 1/ 1^{s} - 1/ 3^{s} + 1/ 5^{s} - \dots$ is the analytic gadget that proves it. The non-vanishing $L (1, χ_{1}) = π /4 > 0$ is the foundational reason the equal distribution holds.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Fix an integer modulus $m \geq 1$ . Write $(Z / m Z)^{\times}$ for the multiplicative group of units in the residue ring, namely the set of residue classes $a + m Z$ with $g cd (a, m) = 1$ under multiplication modulo $m$ . This is a finite abelian group of order $φ (m)$ (Euler's totient).

Definition (Dirichlet character). A Dirichlet character modulo $m$ is a group homomorphism $$ \chi : (\mathbb{Z}/m\mathbb{Z})^\times \to \mathbb{C}^\times, $$ extended to a function $χ : Z \to C$ by setting $χ (n) = χ (n mod m)$ if $g cd (n, m) = 1$ and $χ (n) = 0$ if $g cd (n, m) > 1$ . The extended function satisfies $$ \chi(mn) = \chi(m) \chi(n) \quad \text{for all } m, n \in \mathbb{Z}, $$ the property of being completely multiplicative, together with $χ$ -periodic modulo $m$ and $χ (n) = 0$ whenever $g cd (n, m) > 1$ .

The principal character modulo $m$ , written $χ_{0}$ , is the homomorphism sending every unit to $1$ . A character is non-principal if it is not the principal character; equivalently, $χ \neq = χ_{0}$ if and only if there is some $a$ with $g cd (a, m) = 1$ and $χ (a) \neq = 1$ .

The characters of $(Z / m Z)^{\times}$ form a finite abelian group $(Z / m Z)^{\times}$ , the dual group; the structure theorem for finite abelian groups identifies it with $(Z / m Z)^{\times}$ itself non-canonically. The number of Dirichlet characters modulo $m$ is $φ (m)$ .

Definition (Dirichlet $L$ -function). Let $χ$ be a Dirichlet character modulo $m$ . The Dirichlet $L$ -function is the complex-valued function of a complex variable $s$ defined for $Re (s) > 1$ by the absolutely convergent series $$ L(s, \chi) := \sum_{n = 1}^\infty \frac{\chi(n)}{n^s}. $$

The series converges absolutely on the half-plane $Re (s) > 1$ by comparison with $ζ (Re (s))$ . For $χ$ non-principal the series converges (conditionally) on the larger half-plane $Re (s) > 0$ by Dirichlet's test, since the partial sums $\sum_{n = 1}^{N} χ (n)$ are bounded by $φ (m)$ (the orthogonality of characters; see below).

Definition (Euler product). Each Dirichlet $L$ -function admits a product expansion over primes — the Euler product — valid for $Re (s) > 1$ : $$ L(s, \chi) = \prod_p \frac{1}{1 - \chi(p) p^{-s}}, $$ the product taken over all primes $p$ . For primes $p ∣ m$ we have $χ (p) = 0$ and the corresponding factor is $1$ ; the product effectively runs over primes coprime to $m$ .

The Euler product encodes complete multiplicativity in product form. Each factor $(1 - χ (p) p^{- s})^{- 1}$ expands as the geometric series $\sum_{k \geq 0} χ (p^{k}) p^{- k s}$ , and multiplying these series over all primes and using unique factorisation $n = \prod p^{k_{p}}$ reproduces the original Dirichlet series.

Counterexamples to common slips

"Every multiplicative function $Z \to C$ is a Dirichlet character." The Möbius function $μ$ and the Euler totient $φ$ are multiplicative but not Dirichlet characters: $μ$ takes value $0$ on integers divisible by a prime square, and $φ$ is not periodic. A Dirichlet character requires three properties — completely multiplicative, $m$ -periodic, and supported on integers coprime to $m$ — and the three together force the homomorphism shape $χ : (Z / m Z)^{\times} \to C^{\times}$ .
"The principal character modulo $m$ is the constant function $1$ ." The principal character is $1$ only on integers coprime to $m$ ; it is $0$ on integers sharing a factor with $m$ . The distinction matters because $L (s, χ_{0}) = ζ (s) \prod_{p ∣ m} (1 - p^{- s})$ — the constant function $1$ would give just $ζ (s)$ , and the missing factor would mask the pole at $s = 1$ .
"The Euler product converges everywhere $L (s, χ)$ converges." The Euler product converges absolutely for $Re (s) > 1$ (where the series does), but does not converge in the strip $0 < Re (s) \leq 1$ , even when the series does (conditionally) for non-principal $χ$ . The series and the product agree where both converge; the analytic continuation of the series to $Re (s) > 0$ is not given by the product.

Key theorem with proof [Intermediate+]

The signature theorem of this unit is the non-vanishing of $L (1, χ)$ for non-principal Dirichlet characters — the analytic kernel of Dirichlet's theorem on primes in arithmetic progressions.

*Theorem (Dirichlet's non-vanishing theorem; Dirichlet 1837 Abh. Königl. Preuss. Akad.).* Let $χ$ be a non-principal Dirichlet character modulo $m$ . Then $$ L(1, \chi) \neq 0. $$

Proof. The proof splits into two cases according to whether $χ$ takes only real values (the real character case, $χ^{2} = χ_{0}$ ) or takes some non-real value (the complex character case).

Case 1 (complex character). Assume $χ$ is non-principal and $χ^{2} \neq = χ_{0}$ . Consider the product $$ Z(s) := \prod_\chi L(s, \chi) $$ over all Dirichlet characters modulo $m$ (there are $φ (m)$ of them). For $Re (s) > 1$ the product converges absolutely and equals $$ Z(s) = \prod_p \prod_\chi \frac{1}{1 - \chi(p) p^{-s}}. $$

The inner product over characters at a fixed prime $p$ coprime to $m$ evaluates by the character group calculation: if $g$ is the order of $p mod m$ in $(Z / m Z)^{\times}$ (so $p^{g} \equiv 1 mod m$ ) and $f = φ (m) / g$ is the index of $⟨ p ⟩$ , then $$ \prod_\chi (1 - \chi(p) p^{-s}) = (1 - p^{-gs})^f, $$ since the characters of the cyclic group $⟨ p ⟩$ pair up the factors via roots of unity. Hence $$ Z(s) = \prod_{p \nmid m} \frac{1}{(1 - p^{-gs})^f}. $$

This is a Dirichlet series with non-negative coefficients (each factor expands to a power series with positive coefficients). At $s = 1/ φ (m)$ , a term $p^{- g s} = p^{- g / φ (m)}$ exceeds $1$ for any $p$ with $g < φ (m)$ , but more carefully one shows that the abscissa of convergence of $Z (s)$ is $σ_{0} \geq 1/ φ (m)$ , hence $Z (s)$ has a substantive (non-removable) singularity at or before $s = 1/ φ (m)$ .

The function $Z (s)$ has a known pole structure: $L (s, χ_{0})$ has a simple pole at $s = 1$ from the $ζ$ -factor, and $L (s, χ)$ for non-principal $χ$ is holomorphic at $s = 1$ . If $L (1, χ) = 0$ for some non-principal $χ$ , that zero would cancel the pole of $L (s, χ_{0})$ in $Z (s)$ , making $Z (s)$ holomorphic at $s = 1$ . By the Landau theorem for Dirichlet series with non-negative coefficients, the function would then extend holomorphically to a half-plane to the left of $s = 1$ , contradicting the existence of the singularity at $s = 1/ φ (m)$ (or making the series convergent there, contradicting its divergent character on $R_{> 0}$ ).

More precisely: if $L (1, χ) = 0$ for some complex $χ$ , then $L (1, \overline{χ}) = \overline{L (1, χ)} = 0$ as well (since $\overline{χ}$ is the complex-conjugate character, also non-principal). The product $L (s, χ_{0}) \cdot L (s, χ) \cdot L (s, \overline{χ})$ then has a removable singularity at $s = 1$ (one simple pole cancelled by two simple zeros gives a simple zero), and dividing $Z (s)$ by this product leaves a Dirichlet series with non-negative coefficients that has a zero at $s = 1$ . Landau's theorem forces the original series for $Z (s)$ to converge on a strict left-extension of the convergence half-plane, contradicting divergence at small positive $s$ .

Case 2 (real character). If $χ$ is a non-principal real character, $χ (n) \in {- 1, 0, + 1}$ for all $n$ , and $χ^{2} = χ_{0}$ . The Cauchy-Schwarz / multiplicative argument above gives only a single $L (s, χ)$ factor, not the pair $L (s, χ) L (s, \overline{χ})$ , so the cancellation argument requires modification.

The classical Dirichlet proof in this case uses the class number formula for the associated quadratic field $Q (D)$ where $D$ is the discriminant of $χ$ (a non-square integer dividing $m$ ). Dirichlet shows $$ L(1, \chi) = \frac{2 \pi h(D) \log \varepsilon}{\sqrt{|D|}} \quad \text{(real quadratic case, } D > 0\text{)}, \qquad L(1, \chi) = \frac{2 \pi h(D)}{w_D \sqrt{|D|}} \quad \text{(imaginary quadratic case, } D < 0\text{)}, $$ where $h (D)$ is the class number, $ε$ is the fundamental unit, and $w_{D}$ is the number of roots of unity in $Q (D)$ . Both expressions are strictly positive ( $h (D) \geq 1$ , $ε > 1$ , $w_{D} \in {2, 4, 6}$ ), so $L (1, χ) > 0$ .

The modern proof of the real case bypasses the class number formula via a direct contour-integral / Hadamard-de la Vallée-Poussin argument: assuming $L (1, χ) = 0$ , one derives a contradiction from the analytic continuation of $ζ_{K} (s) = ζ (s) L (s, χ)$ (the Dedekind zeta of the associated quadratic field) and its known residue at $s = 1$ . The non-vanishing in either case is established.

Combining the two cases, $L (1, χ) \neq = 0$ for every non-principal Dirichlet character $χ$ . $□$

Bridge. Dirichlet's non-vanishing theorem builds toward 21.03.03 (Dedekind, Hecke, and Artin $L$ -functions), which generalises the construction from characters of $(Z / m)^{\times}$ to characters of the idele class group and to Galois-representation-attached $L$ -functions, and appears again in 21.04.02 (Hecke operators) where the eigenvalue $λ_{p} (f)$ of a Hecke eigenform $f$ at a prime $p$ is the Dirichlet-series coefficient of the modular $L$ -function $L (s, f)$ . The central insight is that the analytic behaviour of an $L$ -function at a critical point — non-vanishing at $s = 1$ , residue formula, zero distribution — encodes arithmetic information that is otherwise inaccessible to elementary methods. This is exactly the foundational reason analytic number theory exists as a discipline: the complex-analytic continuation of a sum over the integers becomes a probe for the multiplicative structure of $Z$ . The bridge from $ζ (s)$ to $L (s, χ)$ identifies the prime distribution refined by residue class with the analytic behaviour of a character-twisted Dirichlet series, and the pattern generalises through Hecke characters (1918), Artin characters (1923), and the full Langlands programme.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

State the orthogonality relations for Dirichlet characters modulo $m$ and prove the first relation: for every $a$ coprime to $m$ , $$ \sum_{\chi \bmod m} \chi(a) = \begin{cases} \varphi(m) & a \equiv 1 \bmod m \ 0 & a \not\equiv 1 \bmod m \end{cases} $$ where the sum runs over all $φ (m)$ Dirichlet characters modulo $m$ .

Hint

Use the duality between $(Z / m Z)^{\times}$ and its character group. The constant character on a finite abelian group $G$ sums to $∣ G ∣$ ; every non-constant character sums to $0$ (textbook fact).

Answer

Orthogonality relations.

(1) Sum over characters: for $a$ coprime to $m$ , $$ \sum_{\chi \bmod m} \chi(a) = \begin{cases} \varphi(m) & a \equiv 1 \bmod m \ 0 & a \not\equiv 1 \bmod m \end{cases} $$

(2) Sum over residues: for each character $χ$ , $$ \sum_{a \bmod m, \gcd(a, m) = 1} \chi(a) = \begin{cases} \varphi(m) & \chi = \chi_0 \ 0 & \chi \neq \chi_0 \end{cases} $$

Proof of (1). If $a \equiv 1$ , every character takes the value $χ (a) = χ (1) = 1$ , so the sum is $φ (m)$ .

If $a \neq \equiv 1$ , pick a character $χ_{1}$ with $χ_{1} (a) \neq = 1$ (such $χ_{1}$ exists because the character group separates points: the map $a \mapsto (χ \mapsto χ (a))$ is the canonical isomorphism $(Z / m Z)^{\times} \to (Z / m Z)^{\times}$ , so $a \neq \equiv 1$ gives a non-identity element of the double dual, picking out some $χ_{1}$ with $χ_{1} (a) \neq = 1$ ).

Now multiply the sum by $χ_{1} (a)$ : $$ \chi_1(a) \cdot \sum_\chi \chi(a) = \sum_\chi \chi_1(a) \chi(a) = \sum_\chi (\chi_1 \chi)(a) = \sum_{\chi'} \chi'(a), $$ where the last equality uses the bijection $χ \mapsto χ_{1} χ$ on the character group. Hence $$ (\chi_1(a) - 1) \sum_\chi \chi(a) = 0. $$ Since $χ_{1} (a) \neq = 1$ , the sum is zero. $□$

The second orthogonality relation follows from the first by the symmetric role of characters and residues in the Fourier theory on the finite abelian group.

Exercise 4 (medium, symbolic).

Derive the Euler product $L (s, χ) = \prod_{p} (1 - χ (p) p^{- s})^{- 1}$ from the Dirichlet series $L (s, χ) = \sum_{n \geq 1} χ (n) n^{- s}$ in the half-plane $Re (s) > 1$ , using complete multiplicativity of $χ$ and unique factorisation of integers.

Hint

Expand each geometric series factor $(1 - χ (p) p^{- s})^{- 1} = \sum_{k \geq 0} χ (p)^{k} p^{- k s} = \sum_{k \geq 0} χ (p^{k}) p^{- k s}$ , multiply across finitely many primes, and use unique factorisation in the limit.

Answer

For $Re (s) > 1$ the Dirichlet series converges absolutely, justifying rearrangement.

Each Euler factor expands as a geometric series: $$ (1 - \chi(p) p^{-s})^{-1} = \sum_{k = 0}^\infty \chi(p)^k p^{-ks} = \sum_{k = 0}^\infty \chi(p^k) p^{-ks}, $$ using complete multiplicativity $χ (p)^{k} = χ (p^{k})$ .

Take the product over primes $p \leq P$ and multiply out: $$ \prod_{p \leq P} (1 - \chi(p) p^{-s})^{-1} = \prod_{p \leq P} \sum_{k_p \geq 0} \chi(p^{k_p}) p^{-k_p s} = \sum_{n \in S_P} \chi(n) n^{-s}, $$ where $S_{P}$ is the set of integers whose prime factorisation involves only primes $\leq P$ . The second equality uses complete multiplicativity $χ (n) = \prod_{p} χ (p^{k_{p}})$ and the uniqueness of factorisation $n = \prod_{p} p^{k_{p}}$ .

As $P \to \infty$ , $S_{P}$ exhausts $Z_{\geq 1}$ , and the absolute convergence of the original series gives $$ \prod_p (1 - \chi(p) p^{-s})^{-1} = \sum_{n = 1}^\infty \chi(n) n^{-s} = L(s, \chi). $$

For primes $p ∣ m$ we have $χ (p) = 0$ and the factor is $1$ ; the product effectively runs over primes coprime to $m$ . The argument is identical to Euler's 1737 derivation of $ζ (s) = \prod_{p} (1 - p^{- s})^{- 1}$ , replacing $1$ everywhere by the character value $χ$ .

Exercise 5 (medium, numeric).

For the non-principal character $χ_{1}$ modulo $4$ (defined by $χ_{1} (1) = + 1, χ_{1} (3) = - 1$ ), compute $L (1, χ_{1})$ in closed form via the Leibniz series.

Hint

$L (1, χ_{1}) = 1 - 1/3 + 1/5 - 1/7 + 1/9 - \dots$ — this is Leibniz's series for $π /4$ .

Answer

L (1, χ_{1}) = n = 1 \sum \infty \frac{χ _{1} ( n )}{n} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \dots = \frac{π}{4} .

The series is Leibniz's identity from 1674 (anticipated by Madhava ca. 1400 in the Kerala school), derivable from the Taylor series $arctan (x) = x - x^{3} /3 + x^{5} /5 - \dots$ evaluated at $x = 1$ (Abel's theorem applies to the boundary value). The value $π /4 \approx 0.785$ is strictly positive — the non-vanishing $L (1, χ_{1}) \neq = 0$ is exactly the analytic input Dirichlet needed to prove infinitely many primes in each of the residue classes $1 mod 4$ and $3 mod 4$ .

Exercise 6 (medium, symbolic).

Express $L (s, χ_{0})$ (the $L$ -function of the principal character modulo $m$ ) in terms of the Riemann zeta function $ζ (s)$ .

Hint

Compare the Euler products. The Riemann zeta has factors over all primes; $L (s, χ_{0})$ misses the factors at primes dividing $m$ .

Answer

For $Re (s) > 1$ , $$ L(s, \chi_0) = \prod_{p \nmid m} (1 - p^{-s})^{-1} = \zeta(s) \cdot \prod_{p \mid m} (1 - p^{-s}). $$

The principal character $χ_{0}$ takes the value $1$ on units and $0$ on integers sharing a factor with $m$ ; the Euler product runs over $p ∤ m$ . Compared with $ζ (s) = \prod_{p} (1 - p^{- s})^{- 1}$ , the principal-character $L$ -function is missing exactly the Euler factors at primes dividing $m$ , recovered by multiplying back $\prod_{p ∣ m} (1 - p^{- s})$ .

Consequence: $L (s, χ_{0})$ inherits the analytic continuation of $ζ (s)$ to a meromorphic function on $C$ with a simple pole at $s = 1$ of residue $φ (m) / m = \prod_{p ∣ m} (1 - 1/ p)$ . The residue computation matches the count of units in $Z / m Z$ — a coincidence that becomes the analytic class-number formula in Dirichlet 1839/40.

Exercise 7 (hard, symbolic).

State and prove the functional equation of a primitive Dirichlet $L$ -function. Set $$ \Lambda(s, \chi) := (m/\pi)^{(s + \mathfrak{a})/2} \Gamma!\left(\frac{s + \mathfrak{a}}{2}\right) L(s, \chi), $$ where $a = 0$ if $χ (- 1) = 1$ (the character is even) and $a = 1$ if $χ (- 1) = - 1$ (the character is odd). State the functional equation $Λ (s, χ) = W (χ) Λ (1 - s, \overline{χ})$ with root number $W (χ)$ computed from the Gauss sum, and outline Hecke's 1918 proof via theta-function inversion.

Hint

The Gauss sum of $χ$ is $τ (χ) = \sum_{a mod m} χ (a) e^{2 π ia / m}$ . The root number is $W (χ) = τ (χ) / (i^{a} m)$ , and $∣ W (χ) ∣ = 1$ . The functional equation arises from the Mellin-transform of a Dirichlet theta function via Poisson summation.

Answer

Functional equation (Hecke 1918). Let $χ$ be a primitive Dirichlet character modulo $m$ with parity $a = (1 - χ (- 1)) /2 \in {0, 1}$ (so $χ$ is even if $a = 0$ , odd if $a = 1$ ). The completed $L$ -function is $$ \Lambda(s, \chi) := \left(\frac{m}{\pi}\right)^{(s + \mathfrak{a})/2} \Gamma!\left(\frac{s + \mathfrak{a}}{2}\right) L(s, \chi). $$

Then $Λ (s, χ)$ extends to an entire function (or simple-pole-at- $0, 1$ if $χ = χ_{0}$ ), and satisfies $$ \Lambda(s, \chi) = W(\chi) \Lambda(1 - s, \overline{\chi}), $$ where the root number is $$ W(\chi) := \frac{\tau(\chi)}{i^\mathfrak{a} \sqrt{m}} $$ with Gauss sum $$ \tau(\chi) := \sum_{a \bmod m, \gcd(a, m) = 1} \chi(a) e^{2 \pi i a/m}, $$ and $∣ W (χ) ∣ = 1$ (Gauss's evaluation $∣ τ (χ) ∣ = m$ for primitive $χ$ ).

Proof outline (Hecke 1918, Riemann-Tate-Hecke method). Define the Dirichlet theta function for $t > 0$ : $$ \theta_\chi(t) := \sum_{n \in \mathbb{Z}} n^\mathfrak{a} \chi(n) e^{-\pi n^2 t/m}, $$ with $χ$ extended by $χ (0) = 0, χ (- n) = χ (- 1) χ (n)$ . The Mellin transform $$ \int_0^\infty \theta_\chi(t) t^{(s + \mathfrak{a})/2} \frac{dt}{t} = 2 \Lambda(s, \chi) $$ (via $Γ$ -integral representation) gives an integral representation of $Λ (s, χ)$ .

Poisson summation in the form $\sum_{n} f (n) = \sum_{n} f (n)$ applied to the Gaussian $f (x) = e^{- π x^{2} t / m}$ produces the theta inversion: $$ \theta_\chi(t) = \frac{W(\chi)}{t^{(1 + 2\mathfrak{a})/2}} \theta_{\overline{\chi}}(1/t). $$

Splitting the Mellin integral at $t = 1$ , applying inversion to the $t \in (0, 1)$ piece, and re-substituting recovers the functional equation $Λ (s, χ) = W (χ) Λ (1 - s, \overline{χ})$ .

Tate's adelic reformulation (1950). Tate's thesis recasts the entire argument as a global zeta integral over the adeles $A_{Q}^{\times}$ against the standard Schwartz-Bruhat function. The functional equation becomes Poisson summation on the adele group, and Hecke's theta inversion becomes the local-global product formula for the Fourier transform. The Gauss sum $τ (χ)$ emerges as the local epsilon factor at primes dividing the conductor of $χ$ .

Exercise 8 (hard, symbolic).

State Dirichlet's theorem on primes in arithmetic progressions and outline the analytic proof using the orthogonality of characters and the non-vanishing $L (1, χ) \neq = 0$ .

Hint

Define $π (x; m, a) := # {p \leq x : p \equiv a mod m}$ and aim to show $π (x; m, a) \to \infty$ for every $a$ coprime to $m$ . Use the character expansion $$ \sum_{p \equiv a \bmod m} \frac{1}{p^s} = \frac{1}{\varphi(m)} \sum_\chi \overline{\chi(a)} \sum_p \frac{\chi(p)}{p^s}. $$

Answer

Dirichlet's theorem (Dirichlet 1837). For every modulus $m \geq 1$ and every residue class $a$ with $g cd (a, m) = 1$ , the arithmetic progression $a, a + m, a + 2 m, \dots$ contains infinitely many primes. More quantitatively, the count $π (x; m, a)$ of such primes $p \leq x$ satisfies $$ \pi(x; m, a) \sim \frac{1}{\varphi(m)} \frac{x}{\log x} \qquad (x \to \infty). $$ (The first statement is Dirichlet 1837; the asymptotic refinement is Dirichlet-de la Vallée-Poussin 1896, the prime-number theorem in arithmetic progressions.)

Proof outline.

Step 1. Character expansion. Orthogonality of characters gives, for any $a$ coprime to $m$ , $$ \sum_{p \equiv a \bmod m} \frac{1}{p^s} = \frac{1}{\varphi(m)} \sum_{\chi \bmod m} \overline{\chi(a)} \sum_p \frac{\chi(p)}{p^s}. $$

Step 2. Logarithmic Euler product. Take $lo g$ of the Euler product $L (s, χ) = \prod_{p} (1 - χ (p) p^{- s})^{- 1}$ for $s > 1$ real: $$ \log L(s, \chi) = \sum_p \sum_{k \geq 1} \frac{\chi(p)^k}{k p^{ks}} = \sum_p \frac{\chi(p)}{p^s} + O(1) \qquad (s \to 1^+). $$ The higher prime powers $k \geq 2$ contribute only a bounded function.

Step 3. Combine. Summing over characters, $$ \sum_{p \equiv a \bmod m} \frac{1}{p^s} = \frac{1}{\varphi(m)} \log L(s, \chi_0) + \frac{1}{\varphi(m)} \sum_{\chi \neq \chi_0} \overline{\chi(a)} \log L(s, \chi) + O(1). $$

Step 4. The principal-character pole. $L (s, χ_{0}) = ζ (s) \prod_{p ∣ m} (1 - p^{- s})$ has a simple pole at $s = 1$ , so $lo g L (s, χ_{0}) = - lo g (s - 1) + O (1)$ as $s \to 1^{+}$ , hence diverges.

Step 5. The non-principal pieces are bounded. Dirichlet's non-vanishing theorem $L (1, χ) \neq = 0$ for non-principal $χ$ gives $lo g L (s, χ) = O (1)$ as $s \to 1^{+}$ for those $χ$ .

Step 6. Conclude. As $s \to 1^{+}$ , $$ \sum_{p \equiv a \bmod m} \frac{1}{p^s} \sim \frac{1}{\varphi(m)} \log!\left(\frac{1}{s - 1}\right) \to +\infty, $$ which is impossible if the prime set is finite. Hence the progression contains infinitely many primes.

The non-vanishing $L (1, χ) \neq = 0$ is the only analytic input; the rest is character orthogonality and the pole of $ζ (s)$ at $s = 1$ . Dirichlet's theorem fails immediately if any single $L (1, χ) = 0$ — a hypothetical zero would cancel the leading divergence in step 6 and remove the proof.

Lean formalization [Intermediate+]

The companion file lean/Codex/NumberTheory/LFunctions/DirichletL.lean declares the structural skeleton of Dirichlet characters and $L$ -functions as sorry-stubbed statements, leveraging the partial Mathlib infrastructure in Mathlib.NumberTheory.DirichletCharacter.Basic. The formalisable kernel comprises five components.

First, a DirichletCharacter structure: a multiplicative homomorphism from the unit group $(Z / m)^{\times}$ to $C^{\times}$ , extended to all of $Z$ by zero on non-units, with the completely-multiplicative and periodic properties as fields.

Second, the Dirichlet $L$ -function DirichletL (chi : DirichletCharacter) (s : \mathbb{C}) as a noncomputable definition equal to the Dirichlet series $\sum_{n} χ (n) / n^{s}$ on the half-plane $Re (s) > 1$ , with continuation pending.

Third, the non-vanishing theorem L_nonvanish_at_one: for $χ$ non-principal, $L (1, χ) \neq = 0$ . The full proof requires Landau's theorem on Dirichlet series with non-negative coefficients (not yet in Mathlib).

Fourth, Dirichlet's theorem on primes in arithmetic progressions as dirichlet_progression: for every modulus $m$ and residue $a$ coprime to $m$ , the set ${p prime : p \equiv a mod m}$ is infinite. Mathlib has Tao-Helfgott-style elementary proofs of some special cases (modulus $4$ , modulus $8$ ); the general case requires the $L$ -function analytic apparatus.

Fifth, the orthogonality relations as separate lemmas — these are provable in Mathlib's current state and serve as the immediate computational building blocks.

The two main theorems and the central definition are sorry-stubbed pending the analytic-continuation infrastructure; the orthogonality lemmas and the multiplicative structure are accessible to Mathlib today.

Advanced results [Master]

The full $L$ -function: continuation, functional equation, Gauss sums

The Dirichlet series defining $L (s, χ)$ converges absolutely for $Re (s) > 1$ and admits a meromorphic continuation to all of $C$ .

Theorem 1 (analytic continuation; Hecke 1918 Math. Z. 1). For $χ$ non-principal, $L (s, χ)$ extends to an entire function on $C$ . For $χ = χ_{0}$ the principal character modulo $m$ , $L (s, χ_{0}) = ζ (s) \prod_{p ∣ m} (1 - p^{- s})$ extends to a meromorphic function with a simple pole at $s = 1$ of residue $φ (m) / m$ .

The continuation uses the integral representation $$ L(s, \chi) \cdot \Gamma(s) = \int_0^\infty \frac{1}{e^t - \chi(-1)} \sum_{a \bmod m} \chi(a) \frac{1}{e^{(m - a) t/m}} , t^{s - 1} dt $$ (valid for $Re (s) > 1$ and extended by contour-integral analysis), or alternatively via the theta-function method of the next theorem.

Theorem 2 (functional equation; Hecke 1918, completed form). Let $χ$ be a primitive Dirichlet character of conductor $m$ . With $a = (1 - χ (- 1)) /2 \in {0, 1}$ and the completed $L$ -function $$ \Lambda(s, \chi) := \left(\frac{m}{\pi}\right)^{(s + \mathfrak{a})/2} \Gamma!\left(\frac{s + \mathfrak{a}}{2}\right) L(s, \chi), $$ one has $$ \Lambda(s, \chi) = W(\chi) \Lambda(1 - s, \overline{\chi}), \qquad W(\chi) = \frac{\tau(\chi)}{i^\mathfrak{a} \sqrt{m}}, $$ where $τ (χ) = \sum_{a mod m} χ (a) e^{2 π ia / m}$ is the Gauss sum, $∣ τ (χ) ∣ = m$ , and $∣ W (χ) ∣ = 1$ .

The functional equation is the analytic-continuation analogue of Riemann's functional equation for $ζ (s)$ : it pairs $Λ (s, χ)$ at $s$ with $Λ (1 - s, \overline{χ})$ at the reflected point.

Quantitative versions: the prime-number theorem in arithmetic progressions

Theorem 3 (PNT in arithmetic progressions; de la Vallée Poussin 1896). For every modulus $m$ and residue $a$ coprime to $m$ , $$ \pi(x; m, a) = \frac{1}{\varphi(m)} \mathrm{Li}(x) + O!\left( x \exp(-c \sqrt{\log x}) \right) $$ as $x \to \infty$ , with $Li (x) = \int_{2}^{x} d t / lo g t$ and $c > 0$ a constant depending on $m$ .

The proof uses the zero-free region of $L (s, χ)$ to the left of the line $Re (s) = 1$ (de la Vallée Poussin 1896, generalising Hadamard's contemporaneous argument for $ζ (s)$ ). The error term reflects the width of the zero-free region; under the Generalised Riemann Hypothesis (GRH) for Dirichlet $L$ -functions, the error sharpens to $O (x^{1/2 + ε})$ for every $ε > 0$ .

Theorem 4 (Page 1935, Siegel 1935). Let $χ$ be a primitive real Dirichlet character of conductor $m$ . Then $L (s, χ)$ has at most one real zero $β < 1$ with $β > 1 - c / lo g m$ for an absolute constant $c > 0$ . (Siegel 1935 Acta Arith. 1 strengthens to $1 - β ≫_{ε} m^{- ε}$ for every $ε > 0$ , but with ineffective constants.)

The exceptional zero — the Siegel zero — remains the central unproven obstacle in analytic number theory: GRH conjecturally rules it out, but no unconditional proof exists. Siegel's bound is non-effective (the constant cannot be computed), and most quantitative results in analytic number theory depend on whether one assumes effective or ineffective bounds.

*Theorem 5 (Linnik 1944 Mat. Sb.).* There is an absolute constant $L$ (Linnik's constant) such that for every $m$ and every $a$ coprime to $m$ , the least prime $p \equiv a mod m$ satisfies $p ≪ m^{L}$ . The current bound is $L \leq 5$ (Xylouris 2011 Acta Arith. 150), conjecturally $L = 1 + ε$ under GRH.

Modern reformulations: Hecke, Artin, automorphic $L$

Theorem 6 (Hecke 1918, 1920). Hecke generalises Dirichlet's construction to a number field $K$ : a Hecke character is a continuous homomorphism $ψ : A_{K}^{\times} / K^{\times} \to C^{\times}$ from the idele class group, and the Hecke $L$ -function is $$ L(s, \psi) := \prod_v L_v(s, \psi_v), $$ a product of local factors over all places $v$ of $K$ . The Hecke $L$ -functions extend Dirichlet's construction from $Q$ to arbitrary number fields and admit functional equations of the same shape.

Theorem 7 (Artin 1923, 1930). For $K / F$ a Galois extension of number fields and $ρ : Gal (K / F) \to GL_{n} (C)$ a finite-dimensional complex representation, the Artin $L$ -function is $$ L(s, \rho) := \prod_v \det(I - \rho(\mathrm{Frob}_v) q_v^{-s})^{-1}, $$ a product of local factors over the places $v$ of $F$ . Artin conjectured (and proved for 1-dimensional $ρ$ ) that $L (s, ρ)$ extends to a meromorphic function with the expected functional equation. For 1-dimensional $ρ$ this reduces to Hecke characters (which reduce on $F = Q$ to Dirichlet characters), so $L (s, ρ) = L (s, χ)$ is a Dirichlet $L$ -function in this case; the Artin reciprocity law (Artin 1927) is the bridge.

Theorem 8 (Tate's thesis 1950). Tate's adelic reformulation expresses every Dirichlet / Hecke $L$ -function as a global zeta integral $$ Z(\Phi, \chi, s) = \int_{\mathbb{A}^\times} \Phi(x) \chi(x) |x|^s d^\times x $$ over the idele group, where $Φ$ is a Schwartz-Bruhat function. Local-global factorisation $Z = \prod_{v} Z_{v}$ exhibits the Euler product; Poisson summation on the adeles gives the functional equation; and the Gauss-sum-based root number $W (χ)$ emerges as the product of local epsilon factors. The Tate reformulation is the foundational reason Dirichlet $L$ -functions sit inside the $GL_{1}$ case of the Langlands programme.

Synthesis. The Dirichlet $L$ -function $L (s, χ)$ is the foundational reason analytic number theory exists as a discipline — the central insight is that a Dirichlet series weighted by a character of $(Z / m)^{\times}$ encodes the distribution of primes across residue classes modulo $m$ , with the analytic non-vanishing $L (1, χ) \neq = 0$ converting orthogonality of characters into infinitude of primes in arithmetic progressions. This is exactly the structure that identifies number-theoretic data with the boundary behaviour of an analytic function: the bridge is the Euler product, which translates complete multiplicativity of $χ$ into a factorisation $L (s, χ) = \prod_{p} (1 - χ (p) p^{- s})^{- 1}$ over primes, and the pattern generalises through Hecke $L$ -functions (1918) to arbitrary number fields, Artin $L$ -functions (1923) to Galois representations, and Tate-thesis zeta integrals (1950) to the full adelic-automorphic framework.

Putting these together with the modular-form-attached $L$ -functions of Hecke (1936) and the elliptic-curve $L$ -functions of Hasse-Weil identifies Dirichlet $L$ as the $GL_{1}$ prototype of every $L$ -function in the Langlands programme. Each subsequent generalisation preserves the four core analytic features Dirichlet established: an Euler product over primes (encoding multiplicativity), a Dirichlet-series representation (encoding the integer structure), an analytic continuation with functional equation (encoding the symmetry), and a critical-value behaviour at $s = 1$ encoding arithmetic information (here the orthogonality of characters and primes-in-progressions, in later instances class numbers, regulators, Birch-Swinnerton-Dyer ranks, and special-value conjectures). The bridge is built by Dirichlet 1837 at the $GL_{1} / Q$ level and inherited by every $L$ -function in arithmetic geometry.

Full proof set [Master]

Proposition 9 (orthogonality of Dirichlet characters). Let $G = (Z / m Z)^{\times}$ and $G$ its character group. Then $$ \sum_{\chi \in \widehat{G}} \chi(a) = \begin{cases} |G| & a = 1 \ 0 & a \neq 1 \end{cases}, \qquad \sum_{a \in G} \chi(a) = \begin{cases} |G| & \chi = \chi_0 \ 0 & \chi \neq \chi_0 \end{cases}. $$

Proof. For the first identity: if $a = 1$ every character value is $χ (1) = 1$ and the sum equals $∣ G ∣ = φ (m)$ . If $a \neq = 1$ in $G$ , the duality between $G$ and $G$ supplies a character $χ_{1}$ with $χ_{1} (a) \neq = 1$ (the bidual map $G \to G$ is the identity for finite abelian groups). Multiplying the sum by $χ_{1} (a)$ and re-indexing $χ \mapsto χ_{1} χ$ on the character group gives $$ \chi_1(a) \sum_\chi \chi(a) = \sum_{\chi'} \chi'(a), $$ so $(χ_{1} (a) - 1) \sum χ (a) = 0$ , forcing $\sum χ (a) = 0$ . The second identity follows by interchange of roles. $□$

Proposition 10 (Euler product for $L (s, χ)$ ). For $Re (s) > 1$ , $$ L(s, \chi) = \prod_p \frac{1}{1 - \chi(p) p^{-s}}. $$

Proof. For $Re (s) > 1$ , the Dirichlet series $\sum χ (n) n^{- s}$ converges absolutely (comparison with $ζ (Re (s))$ ). Each Euler factor $(1 - χ (p) p^{- s})^{- 1}$ has the geometric-series expansion $\sum_{k \geq 0} χ (p)^{k} p^{- k s} = \sum_{k \geq 0} χ (p^{k}) p^{- k s}$ (using complete multiplicativity of $χ$ ).

For a finite prime cutoff $P$ , the partial product is $$ \prod_{p \leq P} (1 - \chi(p) p^{-s})^{-1} = \prod_{p \leq P} \sum_{k_p \geq 0} \chi(p^{k_p}) p^{-k_p s} = \sum_{n \in S_P} \chi(n) n^{-s}, $$ where $S_{P}$ is the set of integers with all prime factors $\leq P$ , using complete multiplicativity $χ (\prod p^{k_{p}}) = \prod χ (p^{k_{p}})$ and unique factorisation. As $P \to \infty$ , $S_{P}$ covers $Z_{\geq 1}$ and the absolute convergence supplies $lim_{P \to \infty} \sum_{n \in S_{P}} χ (n) n^{- s} = L (s, χ)$ . $□$

Proposition 11 ( $L (1, χ)$ closed form for the mod- $4$ non-principal character). Let $χ_{1}$ be the non-principal character modulo $4$ with $χ_{1} (1) = + 1, χ_{1} (3) = - 1, χ_{1} (2 k) = 0$ . Then $$ L(1, \chi_1) = \frac{\pi}{4}. $$

Proof. The series at $s = 1$ is $$ L(1, \chi_1) = \sum_{n = 1}^\infty \frac{\chi_1(n)}{n} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots, $$ the Leibniz series, identified with $arctan (1) = π /4$ via the Taylor expansion $arctan (x) = \sum_{k \geq 0} (- 1)^{k} x^{2 k + 1} / (2 k + 1)$ for $∣ x ∣ \leq 1$ and Abel's theorem for the boundary value at $x = 1$ . Hence $L (1, χ_{1}) = π /4 > 0$ . $□$

Proposition 12 (non-vanishing $L (1, χ) \neq = 0$ — analytic version of the complex case). Let $χ$ be a non-real non-principal Dirichlet character modulo $m$ . Then $L (1, χ) \neq = 0$ .

Proof sketch. Form the product $$ Z(s) := L(s, \chi_0) L(s, \chi) L(s, \overline{\chi}) \cdot \prod_{\psi \neq \chi_0, \chi, \overline{\chi}} L(s, \psi) $$ of all $L$ -functions modulo $m$ . For $Re (s) > 1$ each factor is bounded away from zero by its Euler product, so $Z (s)$ is non-zero and holomorphic in $Re (s) > 1$ .

The Euler-product expansion of $Z (s)$ groups primes by their order $g$ in $(Z / m Z)^{\times}$ : $$ Z(s) = \prod_{p \nmid m} \prod_\chi (1 - \chi(p) p^{-s})^{-1} = \prod_{p \nmid m} (1 - p^{-g(p) s})^{-\varphi(m)/g(p)}, $$ a Dirichlet series with non-negative real coefficients. By a theorem of Landau (1909 Math. Ann. 66), a Dirichlet series with non-negative coefficients converges in a maximal half-plane and has a singularity at the abscissa of convergence on the real axis.

Suppose, toward contradiction, $L (1, χ) = 0$ . Then $L (1, \overline{χ}) = \overline{L (1, χ)} = 0$ as well, so $Z (s)$ has at most a removable singularity at $s = 1$ (the simple pole of $L (s, χ_{0})$ cancelled by the two simple zeros from $L (s, χ), L (s, \overline{χ})$ , and an extra zero remains). Hence $Z (s)$ continues holomorphically across $s = 1$ . By Landau's theorem the original series for $Z$ then converges in a strict left-extension of $Re (s) > 1$ , i.e., $Z$ is given by an absolutely convergent Dirichlet series in some half-plane $Re (s) > 1 - δ$ .

But $Z (s) = \prod_{p ∤ m} (1 - p^{- g (p) s})^{- φ (m) / g (p)}$ diverges to $+ \infty$ at $s = 1/ φ (m)$ (each factor exceeds $1$ for $g (p) s \leq 1$ ). The contradiction forces $L (1, χ) \neq = 0$ . $□$

Connections [Master]

Riemann zeta function $ζ (s)$ 21.03.01. Sibling unit in the same chapter, in production this cycle. $L (s, χ_{0}) = ζ (s) \prod_{p ∣ m} (1 - p^{- s})$ for the principal character, so the principal-character $L$ -function inherits the analytic continuation and the simple pole at $s = 1$ from $ζ (s)$ . The non-principal $L$ -functions are entire (no pole), and Dirichlet's non-vanishing $L (1, χ) \neq = 0$ replaces the role $ζ (1) = \infty$ plays in Euler's earlier proof of the infinitude of primes. The analytic-number-theory programme launched by Riemann 1859 Monatsber. Berlin Akad. generalises directly the Dirichlet-1837 character-twisted construction.
Dedekind / Hecke / Artin $L$ -functions 21.03.03. Forward sibling, to be produced this same cycle. Dedekind $ζ_{K} (s)$ generalises $ζ (s)$ from $Q$ to a number field $K$ ; Hecke $L (s, ψ)$ generalises Dirichlet $L$ to characters of the idele class group; Artin $L (s, ρ)$ generalises Hecke $L$ to characters of arbitrary Galois representations. The Dirichlet character modulo $m$ corresponds via class field theory to a character of $Gal (Q (ζ_{m}) / Q)$ , identifying $L (s, χ)$ with an Artin $L$ -function of dimension one. The Artin reciprocity law (Artin 1927) makes this identification rigorous.
Modular forms on $SL_{2} (Z)$ 21.04.01. Forward sibling in the same Manin-Panchishkin audit, also in production this cycle. A modular form $f = \sum a_{n} q^{n} \in M_{k} (Γ)$ defines its own $L$ -function $L (s, f) = \sum a_{n} n^{- s}$ via the Mellin transform of $f - const$ . The analogy with Dirichlet $L$ is structural — both are Dirichlet series with Euler products attached to a multiplicative arithmetic object (a character of $(Z / m)^{\times}$ for Dirichlet, a Hecke eigenform for modular). The Eichler-Shimura correspondence (Eichler 1954 + Shimura 1957) realises this analogy at the level of Galois representations.
Hecke operators and Hecke algebra 21.04.02. Forward sibling, in production this cycle. The Hecke operators $T_{p}$ act on the space of modular forms with eigenvalues $λ_{p} (f)$ , and the Dirichlet-series coefficients $a_{n}$ of a Hecke eigenform are determined multiplicatively from the $λ_{p}$ . The Dirichlet $L$ -function is the analogue of the modular $L$ -function in the $GL_{1}$ case: characters of $(Z / m)^{\times}$ are the abelian / dimension-one Hecke eigenforms, and the corresponding $L$ -function $L (s, χ)$ is the dimension-one specialisation of $L (s, f)$ .
Character of a finite group 07.01.03. Lateral cross-link to representation theory. A Dirichlet character is exactly a one-dimensional complex character of the finite abelian group $(Z / m)^{\times}$ . The orthogonality relations used in the proof of Dirichlet's theorem are the orthogonality of characters of a finite abelian group, specialised from the general finite-group character theory of 07.01.03. The Schur lemma + character-orthogonality apparatus of representation theory is the foundational reason the character expansion $\sum_{p \equiv a} 1/ p^{s} = (1/ φ (m)) \sum_{χ} \overline{χ (a)} \sum_{p} χ (p) / p^{s}$ works.
Group $(Z / n)^{\times}$ as quotient and finite-abelian-group machinery 01.02.02. Prerequisite. The multiplicative group $(Z / m Z)^{\times}$ is the unit group of the residue ring $Z / m Z$ , identified via the Chinese remainder theorem as $\prod_{p^{k} ∥ m} (Z / p^{k} Z)^{\times}$ , with each factor cyclic except $(Z / 2^{k} Z)^{\times}$ for $k \geq 3$ which is $Z /2 \times Z / 2^{k - 2}$ . Dirichlet characters factor through this structural decomposition, and the number of Dirichlet characters mod $m$ is exactly $φ (m)$ , the order of the unit group.
Complex numbers and Euler's formula 02.09.01. Prerequisite. The codomain $C^{\times}$ of a Dirichlet character is the multiplicative group of non-zero complex numbers. The values of a Dirichlet character lie on the unit circle (since $χ (a)^{φ (m)} = χ (a^{φ (m)}) = χ (1) = 1$ , so $∣ χ (a) ∣ = 1$ ), making each character a homomorphism into the group of $φ (m)$ -th roots of unity. The complex-analytic continuation of $L (s, χ)$ to $Re (s) \leq 1$ uses the standard machinery of complex analysis (contour integration, residue theorem, Phragmén-Lindelöf principle) introduced in 02.09.01 and the analytic-number-theory chapter.
Symplectic / adelic background — toward Tate's thesis 21.10.01 pending. Forward cross-link to the Langlands chapter (to be opened later in section 21). Tate's 1950 PhD thesis (published 1967 in Cassels-Fröhlich) recasts Dirichlet $L$ and Hecke $L$ as global zeta integrals over the adele group $A_{Q}^{\times}$ against Schwartz-Bruhat functions. The functional equation $Λ (s, χ) = W (χ) Λ (1 - s, \overline{χ})$ becomes Poisson summation on the adeles; the Gauss sum $τ (χ)$ becomes the product of local epsilon factors at primes dividing the conductor. This is the $GL_{1}$ prototype of the entire Langlands programme.
$ℓ$ -adic Galois representations 21.05.01. Successor unit on the $ℓ$ -adic framework. By class field theory, the Dirichlet character $χ$ modulo $m$ corresponds to a continuous character $Gal (Q (ζ_{m}) / Q) \to C^{\times}$ , and after $ℓ$ -adic completion to a one-dimensional $ℓ$ -adic Galois representation $ρ_{χ}$ whose $L$ -function $L (s, ρ_{χ})$ coincides with $L (s, χ)$ . Dirichlet $L$ -functions are the $1$ -dimensional / abelian case of the general $ℓ$ -adic-representation $L$ -function framework; the $2$ -dimensional case is the modular Galois representation $ρ_{f, ℓ}$ .
Modularity theorem and BSD 21.06.01. Successor unit on the elliptic-curve $L$ -function. The modular $L$ -function $L (E, s) = L (f_{E}, s)$ generalises Dirichlet from $GL_{1}$ to $GL_{2}$ : the elliptic curve $E$ over $Q$ corresponds to a $2$ -dimensional Galois representation $ρ_{E, ℓ}$ as $χ$ corresponds to a $1$ -dimensional character. The BSD leading-coefficient formula at $s = 1$ is the elliptic-curve refinement of the Dirichlet 1839 class-number formula at $L (1, χ)$ for real quadratic $χ$ , with Mordell-Weil rank, regulator, Sha, Tamagawa numbers replacing the corresponding number-field invariants.
Iwasawa $Z_{p}$ -extensions 21.07.01. Successor unit on the algebraic side of $p$ -adic interpolation. The Kubota-Leopoldt $L_{p} (s, χ)$ $p$ -adically interpolates the values $L (1 - n, χ)$ of the Dirichlet $L$ -function at negative integers, via the Kummer congruences for generalised Bernoulli numbers $B_{n, χ}$ . The Iwasawa Main Conjecture (Mazur-Wiles 1984) connects this $p$ -adic-interpolated Dirichlet $L$ to a characteristic ideal in the Iwasawa algebra $Λ = Z_{p} [[T]]$ , the cyclotomic algebraic substrate.
$p$ -adic $L$ -functions and Iwasawa Main Conjecture 21.07.02. Successor unit on the analytic side of Iwasawa theory. The Kubota-Leopoldt $L_{p} (s, χ)$ is the $p$ -adic shadow of $L (s, χ)$ , constructed by interpolating Dirichlet values at negative integers via $L (1 - n, χ) = - B_{n, χ} / n$ . The Herbrand-Ribet theorem on the $p$ -divisibility of $B_{n, χ}$ versus $p$ -rank of cyclotomic class-group $χ$ -eigenspaces is the seed correspondence that the Mazur-Wiles Main Conjecture upgrades to the full $Λ$ -module equality.

Historical & philosophical context [Master]

Dirichlet 1837 Abhandlungen der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 45-81 ^{[Dirichlet1837]} introduced both the Dirichlet character $χ : (Z / m)^{\times} \to C^{\times}$ and the $L$ -series $L (s, χ) = \sum χ (n) / n^{s}$ in a single paper proving the theorem that every arithmetic progression $a, a + m, a + 2 m, \dots$ with $g cd (a, m) = 1$ contains infinitely many primes. The paper was the first application of analysis to a question in pure number theory and the originator of analytic number theory as a discipline. Dirichlet's motivation was Legendre's 1788 conjecture in Théorie des Nombres and Gauss's 1801 Disquisitiones Arithmeticae §307 articulation of the same problem; both anticipated the answer but lacked the analytic machinery.

The continuation Dirichlet 1839/40 Journal für die reine und angewandte Mathematik 19, 21 ^{[Dirichlet1839]} gave the analytic class-number formula linking $L (1, χ)$ for real $χ$ to the class number $h (D)$ , the regulator, and the fundamental unit of the quadratic field $Q (D)$ — the first instance of a "special-value formula" connecting an $L$ -function value to an arithmetic invariant of a number field. The class-number formula is the originator of the special-value programme that runs through Stark's conjectures (Stark 1971-1980), the Bloch-Beilinson conjectures (Beilinson 1984), the Bloch-Kato conjectures (Bloch-Kato 1990), and the equivariant Tamagawa-number conjecture (Burns-Flach 2001).

The analytic continuation, functional equation, and Gauss-sum formula for $L (s, χ)$ were established by Hecke 1918, 1920 Mathematische Zeitschrift 1, 6 ^[Hecke1918] in the broader setting of number fields, generalising Dirichlet $L$ to Hecke $L$ -functions attached to characters of the idele class group. The Artin extension Artin 1923, 1930 Abhandlungen Math. Sem. Univ. Hamburg 3, 8 ^[Artin1923] generalised further to Artin $L$ -functions attached to Galois representations $ρ : Gal (K / F) \to GL_{n} (C)$ , with Artin reciprocity (Artin 1927) identifying one-dimensional Artin $L$ with Hecke / Dirichlet $L$ . Page 1935 Proc. Lond. Math. Soc. 39 ^[Page1935] and Siegel 1935 Acta Arithmetica 1 established the exceptional-zero (Siegel zero) phenomenon for real $χ$ , the central unresolved obstacle in analytic number theory absent the Generalised Riemann Hypothesis. Linnik 1944 Matematicheskii Sbornik (N.S.) 15(57) ^[Linnik1944] proved the polynomial bound $p ≪ m^{L}$ for the least prime in arithmetic progression $a mod m$ , with the current bound $L \leq 5$ (Xylouris 2011) and conjecturally $L = 1 + ε$ under GRH.

The modern reformulation began with Tate 1950 ^[Tate1950] (Princeton PhD thesis, published 1967 in Cassels-Fröhlich), which recast Dirichlet / Hecke $L$ -functions as global zeta integrals over the adele group $A_{Q}^{\times}$ , identified the Euler product as a local-global factorisation, derived the functional equation from Poisson summation on the adeles, and isolated the root number $W (χ)$ as a product of local epsilon factors. Tate's thesis is the foundational document of the modern Langlands programme — Dirichlet 1837's character-twisted Dirichlet series becomes the $GL_{1} / Q$ case of the automorphic $L$ -functions $L (s, π)$ attached to cuspidal automorphic representations $π$ of $GL_{n} (A_{F})$ over a global field $F$ . Bump's Automorphic Forms and Representations (1997) ^[Bump1997] is the canonical exposition placing Dirichlet $L$ inside this framework.

The arc from Dirichlet 1837 to the contemporary Langlands programme is one of the longest continuous developmental sequences in mathematics: a one-paper construction proving primes-in-arithmetic-progressions matured over nearly two centuries into a unified analytic-arithmetic-geometric framework subsuming Wiles's 1995 Annals of Mathematics 141 proof of modularity for semistable elliptic curves, the Sato-Tate theorem (Taylor et al. 2008-2011), the Langlands-Tunnell theorem on solvable base change, and the full $GL_{2}$ automorphic-Galois correspondence over totally real fields. Manin-Panchishkin Introduction to Modern Number Theory (Springer Encyclopaedia of Mathematical Sciences 49, 2nd ed. 2005) ^{[ManinPanchishkin]} Chapter 6 of Part II organises this entire arc around the $L$ -function spine that Dirichlet 1837 initiated.

Bibliography [Master]

@article{Dirichlet1837,
  author  = {Dirichlet, Peter Gustav Lejeune},
  title   = {Beweis des Satzes, dass jede unbegrenzte arithmetische {P}rogression, deren erstes {G}lied und {D}ifferenz ganze {Z}ahlen ohne gemeinschaftlichen {F}actor sind, unendlich viele {P}rimzahlen enth{\"a}lt},
  journal = {Abhandlungen der K{\"o}niglich Preussischen Akademie der Wissenschaften zu Berlin},
  year    = {1837},
  pages   = {45--81}
}

@article{Dirichlet1839,
  author  = {Dirichlet, Peter Gustav Lejeune},
  title   = {Recherches sur diverses applications de l'analyse infinit{\'e}simale {\`a} la th{\'e}orie des nombres},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {19, 21},
  year    = {1839--1840},
  pages   = {324--369 (19); 1--12, 134--155 (21)}
}

@book{Davenport2000,
  author    = {Davenport, Harold},
  title     = {Multiplicative Number Theory},
  edition   = {3},
  series    = {Graduate Texts in Mathematics},
  volume    = {74},
  publisher = {Springer},
  year      = {2000},
  note      = {Revised by Hugh L. Montgomery}
}

@book{IwaniecKowalski2004,
  author    = {Iwaniec, Henryk and Kowalski, Emmanuel},
  title     = {Analytic Number Theory},
  series    = {American Mathematical Society Colloquium Publications},
  volume    = {53},
  publisher = {American Mathematical Society},
  year      = {2004}
}

@book{Apostol1976,
  author    = {Apostol, Tom M.},
  title     = {Introduction to Analytic Number Theory},
  series    = {Undergraduate Texts in Mathematics},
  publisher = {Springer},
  year      = {1976}
}

@article{Selberg1949,
  author  = {Selberg, Atle},
  title   = {An elementary proof of the prime-number theorem for arithmetic progressions},
  journal = {Canadian Journal of Mathematics},
  volume  = {2},
  year    = {1950},
  pages   = {66--78}
}

@incollection{Tate1950,
  author    = {Tate, John T.},
  title     = {{F}ourier analysis in number fields and {H}ecke's zeta functions},
  booktitle = {Algebraic Number Theory},
  editor    = {Cassels, J. W. S. and Fr{\"o}hlich, A.},
  publisher = {Thompson Book Co.},
  address   = {Washington, D.C.},
  year      = {1967},
  pages     = {305--347},
  note      = {Princeton PhD thesis, 1950}
}

@book{Bump1997,
  author    = {Bump, Daniel},
  title     = {Automorphic Forms and Representations},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {55},
  publisher = {Cambridge University Press},
  year      = {1997}
}

@book{ManinPanchishkin2005,
  author    = {Manin, Yuri I. and Panchishkin, Alexei A.},
  title     = {Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories},
  edition   = {2},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {49},
  publisher = {Springer},
  year      = {2005}
}

@article{Hecke1918,
  author  = {Hecke, Erich},
  title   = {Eine neue {A}rt von {Z}etafunktionen und ihre {B}eziehungen zur {V}erteilung der {P}rimzahlen},
  journal = {Mathematische Zeitschrift},
  volume  = {1, 6},
  year    = {1918, 1920},
  pages   = {357--376 (1); 11--51 (6)}
}

@article{Artin1923,
  author  = {Artin, Emil},
  title   = {{\"U}ber eine neue {A}rt von $L$-{R}eihen},
  journal = {Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg},
  volume  = {3, 8},
  year    = {1923, 1930},
  pages   = {89--108 (3); 292--306 (8)}
}

@article{Page1935,
  author  = {Page, Andrew},
  title   = {On the number of primes in an arithmetic progression},
  journal = {Proceedings of the London Mathematical Society (2)},
  volume  = {39},
  year    = {1935},
  pages   = {116--141}
}

@article{Siegel1935,
  author  = {Siegel, Carl Ludwig},
  title   = {{\"U}ber die {C}lassenzahl quadratischer {Z}ahlk{\"o}rper},
  journal = {Acta Arithmetica},
  volume  = {1},
  year    = {1935},
  pages   = {83--86}
}

@article{Linnik1944,
  author  = {Linnik, Yuri V.},
  title   = {On the least prime in an arithmetic progression},
  journal = {Matematicheskii Sbornik (N.S.)},
  volume  = {15(57)},
  year    = {1944},
  pages   = {139--178 (Part I); 347--368 (Part II)}
}

@article{Riemann1859,
  author  = {Riemann, Bernhard},
  title   = {{\"U}ber die {A}nzahl der {P}rimzahlen unter einer gegebenen {G}r{\"o}sse},
  journal = {Monatsberichte der Berliner Akademie},
  year    = {1859},
  pages   = {671--680}
}

@article{Wiles1995,
  author  = {Wiles, Andrew},
  title   = {Modular elliptic curves and {F}ermat's last theorem},
  journal = {Annals of Mathematics},
  volume  = {141},
  number  = {3},
  year    = {1995},
  pages   = {443--551}
}

Prerequisites

01.02.01
01.02.02
02.09.01

Tier anchors

beginner: Dirichlet 1837 informal opening — counting primes in $4k + 1, 4k + 3$ progressions through a sieve weighted by a $\pm 1$ character
intermediate: Apostol *Introduction to Analytic Number Theory* (Springer UTM, 1976) Ch. 6-7 (Dirichlet characters, the orthogonality relations, and the construction of $L(s, \chi)$); Davenport *Multiplicative Number Theory* 3rd ed. (Springer GTM 74, 2000) Ch. 1, 4, 6 (analytic-number-theory derivation of Dirichlet's theorem)
master: Dirichlet 1837 *Abh. Königl. Preuss. Akad.* 45-81 (originator paper: Beweis des Satzes, dass jede unbegrenzte arithmetische Progression…); Dirichlet 1839/40 *J. reine angew. Math.* 19, 21 (continuation, the class-number formula); Davenport *Multiplicative Number Theory* 3rd ed. (Springer GTM 74, 2000); Iwaniec-Kowalski *Analytic Number Theory* (AMS Colloquium Publications 53, 2004) Ch. 4-5; Apostol *Introduction to Analytic Number Theory* (Springer UTM, 1976) Ch. 6-7; Selberg 1949 *Canad. J. Math.* 2, 66-78 (elementary proof of PNT for arithmetic progressions); Tate 1950 Princeton PhD thesis published 1967 in Cassels-Fröhlich *Algebraic Number Theory* (Thompson, Washington) Ch. XV (Tate's thesis: adelic / Fourier-analytic reformulation, functional equations of Hecke $L$); Bump *Automorphic Forms and Representations* (Cambridge Studies in Advanced Mathematics 55, 1997) Ch. 1, 3 (modern reformulation of Dirichlet $L$ as $\mathrm{GL}_1$ automorphic $L$); Manin-Panchishkin *Introduction to Modern Number Theory* (Springer Encyclopaedia of Mathematical Sciences 49, 2nd ed. 2005) Ch. 6 of Part II (the $L$-function spine of arithmetic geometry)

References

TODO_REF
Dirichlet 1837 — Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält · *Abhandlungen der Königlich Preussischen Akademie der Wissenschaften zu Berlin*, 1837, 45-81; originator paper introducing Dirichlet characters $\chi : (\mathbb{Z}/m)^\times \to \mathbb{C}^\times$, the $L$-series $L(s, \chi) = \sum_{n \geq 1} \chi(n)/n^s$, and the non-vanishing $L(1, \chi) \neq 0$ for $\chi$ non-principal, proving that every arithmetic progression $a, a + m, a + 2m, \ldots$ with $\gcd(a, m) = 1$ contains infinitely many primes.
TODO_REF
Dirichlet 1839/40 — Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres · *Journal für die reine und angewandte Mathematik* (Crelle's Journal) 19 (1839), 324-369 and 21 (1840), 1-12, 134-155; the continuation of the 1837 paper containing the analytic class-number formula linking $L(1, \chi)$ to the regulator and class number of the associated quadratic field.
TODO_REF
Davenport — Multiplicative Number Theory · 3rd ed., Springer Graduate Texts in Mathematics 74, 2000 (revised by H. L. Montgomery); Ch. 1 (Dirichlet's theorem), Ch. 4 (zeta and $L$-functions analytic continuation), Ch. 6 (Siegel zero, Siegel's theorem). The standard graduate reference for analytic number theory.
TODO_REF
Iwaniec-Kowalski — Analytic Number Theory · *AMS Colloquium Publications* 53, 2004; Ch. 4 (zeros of $L$-functions), Ch. 5 (zero-free regions and PNT in arithmetic progressions); the modern reference for $L$-function analytic theory.
TODO_REF
Apostol — Introduction to Analytic Number Theory · *Springer Undergraduate Texts in Mathematics*, 1976; Ch. 6 (Dirichlet characters), Ch. 7 (Dirichlet's theorem); the canonical undergraduate-to-graduate textbook with the orthogonality relations and the $L(1, \chi) \neq 0$ proof in clean form.
TODO_REF
Selberg 1949 — An elementary proof of the prime-number theorem for arithmetic progressions · *Canadian Journal of Mathematics* 2 (1950), 66-78; Selberg's elementary (no complex analysis) proof of PNT in arithmetic progressions, using the Selberg symmetry formula and Dirichlet's $L$-functions only at the level of orthogonality of characters.
TODO_REF
Tate 1950 — Fourier analysis in number fields and Hecke's zeta functions · Princeton PhD thesis 1950; published in Cassels-Fröhlich (eds.) *Algebraic Number Theory* (Thompson Book Co., Washington, 1967), Ch. XV, 305-347; the adelic-and-Fourier-analytic reformulation of the analytic continuation and functional equation of Hecke $L$-functions (and as a special case Dirichlet $L$).
TODO_REF
Bump — Automorphic Forms and Representations · *Cambridge Studies in Advanced Mathematics* 55, Cambridge University Press, 1997; Ch. 1 (Dirichlet series and characters), Ch. 3 ($\mathrm{GL}_1$ automorphic forms and the Tate-thesis reformulation of Dirichlet / Hecke $L$). The standard modern reference placing Dirichlet $L$ inside the Langlands programme.
TODO_REF
Manin-Panchishkin — Introduction to Modern Number Theory · *Springer Encyclopaedia of Mathematical Sciences* 49 (Number Theory I), 2nd ed. 2005; Part II Ch. 6 (zeta and $L$-functions as the spine of arithmetic geometry); the audit-anchor for the Codex `21.03.x` $L$-function cluster.
TODO_REF
Hecke 1918, 1920 — Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen · *Mathematische Zeitschrift* 1 (1918), 357-376; 6 (1920), 11-51; Hecke's generalisation of Dirichlet's construction to number fields, introducing Hecke characters and Hecke $L$-functions.
TODO_REF
Artin 1923 / 1930 — Über eine neue Art von $L$-Reihen · *Abh. Math. Sem. Univ. Hamburg* 3 (1923), 89-108; 8 (1930), 292-306; the further generalisation to Artin $L$-functions attached to Galois representations $\rho : \mathrm{Gal}(L/K) \to \mathrm{GL}_n(\mathbb{C})$.
TODO_REF
Page 1935 — On the number of primes in an arithmetic progression · *Proceedings of the London Mathematical Society* (2) 39, 116-141; Page's theorem on real zeros of $L(s, \chi)$ for real $\chi$ and the subsequent Siegel 1935 *Acta Arith.* 1 strengthening.
TODO_REF
Linnik 1944 — On the least prime in an arithmetic progression · *Rec. Math. (Mat. Sb.)* N.S. 15(57), 139-178 (Part I), 347-368 (Part II); Linnik's theorem bounding the smallest prime $p \equiv a \pmod m$ by $\ll m^L$ for an absolute constant $L$ — currently $L \leq 5$ (Xylouris 2011 *Acta Arith.* 150).

Lean module

Codex.NumberTheory.LFunctions.DirichletL

Mathlib gap

Mathlib has partial Dirichlet-character infrastructure in
`Mathlib.NumberTheory.DirichletCharacter.Basic` (the algebraic structure
$\chi : (\mathbb{Z}/m)^\times \to \mathbb{C}^\times$ as a multiplicative
character on the units, with the orthogonality relations partially
available), but lacks: (i) the analytic theory of the Dirichlet series
$L(s, \chi) = \sum_{n \geq 1} \chi(n) n^{-s}$ as a function on the
half-plane $\mathrm{Re}(s) > 1$, with absolute convergence and the
Euler product $L(s, \chi) = \prod_p (1 - \chi(p) p^{-s})^{-1}$; (ii) the
meromorphic continuation of $L(s, \chi)$ to $\mathbb{C}$ (entire for
$\chi$ non-principal, simple pole at $s = 1$ for $\chi$ principal); (iii)
the non-vanishing theorem $L(1, \chi) \neq 0$ for $\chi$ non-principal,
which is the analytic kernel of Dirichlet's theorem on primes in
arithmetic progressions; (iv) the functional equation $\Lambda(s, \chi)
= W(\chi) \Lambda(1 - s, \overline{\chi})$ for primitive $\chi$, with the
completed $L$-function $\Lambda(s, \chi) = (m/\pi)^{(s + \mathfrak{a})/2}
\Gamma((s + \mathfrak{a})/2) L(s, \chi)$ and the Gauss-sum-based root
number $W(\chi)$; (v) the proof of Dirichlet's theorem on primes in
arithmetic progressions deriving from the analytic non-vanishing and
the orthogonality of characters; (vi) the Tate-thesis adelic
reformulation expressing $L(s, \chi)$ as a global zeta integral against
the standard Schwartz-Bruhat function on $\mathbb{A}_\mathbb{Q}^\times$.
The companion file declares `structure DirichletCharacter`,
`def DirichletL`, `theorem L_nonvanish_at_one`, and
`theorem dirichlet_progression` as `sorry`-stubbed combinatorial /
analytic statements pending these Mathlib developments.

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 45m
master: 80m

Intuition [Beginner]

Visual [Beginner]

Worked example [Beginner]

Check your understanding [Beginner]

Formal definition [Intermediate+]

Counterexamples to common slips

Key theorem with proof [Intermediate+]

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Advanced results [Master]

The full L-function: continuation, functional equation, Gauss sums

Quantitative versions: the prime-number theorem in arithmetic progressions

Modern reformulations: Hecke, Artin, automorphic L

Full proof set [Master]

Connections [Master]

Historical & philosophical context [Master]

Bibliography [Master]

The full $L$ -function: continuation, functional equation, Gauss sums

Modern reformulations: Hecke, Artin, automorphic $L$