Dirichlet Series and L-functions — Mathematics Knowledge System

Overview

A Dirichlet series is a series of the form $\sum_{n=1}^\infty a_n n^{-s}$. The Riemann zeta function is the canonical example. L-functions are Dirichlet series with multiplicative coefficients attached to arithmetic objects (characters, elliptic curves, modular forms) and are central tools in analytic number theory.

Intuition

The Riemann zeta function $\zeta(s) = \sum n^{-s}$ encodes all prime information via the Euler product $\prod_p (1-p^{-s})^{-1}$. Zeros of $\zeta$ control the error in the prime counting function $\pi(x)$ — the Riemann Hypothesis asserts all non-trivial zeros lie on the critical line $\text{Re}(s) = 1/2$, which would give the sharpest possible prime number theorem.

Formal Definition

Definition

F(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}, \quad s = \sigma + it \in \mathbb{C}

General Dirichlet series

dirichlet-series

\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \text{Re}(s) > 1

Riemann zeta function with Euler product

riemann-zeta

L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s} = \prod_{p} \frac{1}{1 - \chi(p)p^{-s}}, \quad \text{Re}(s) > 1

Dirichlet L-function for a Dirichlet character chi

dirichlet-L

\pi(x) = \text{Li}(x) - \sum_{\rho} \text{Li}(x^\rho) + O(1)

Explicit formula: primes controlled by zeros rho of zeta

explicit-formula

Notation

Notation	Meaning
$\zeta(s)$	Riemann zeta function
$L(s, \chi)$	Dirichlet L-function attached to character chi
$\chi$	Dirichlet character mod q
$\pi(x)$	Prime counting function: number of primes ≤ x
$\text{Li}(x)$	Logarithmic integral: ∫₂ˣ dt/ln t

Theorems

Theorem 1: Dirichlet's Theorem on Primes in Arithmetic Progressions

If \gcd(a, q) = 1, there are infinitely many primes p \equiv a \pmod{q}. The key ingredient is L(1, \chi) \neq 0 for all non-principal characters \chi.

Theorem 2: Analytic Continuation and Functional Equation of zeta

\zeta(s) extends to a meromorphic function on \mathbb{C} with a simple pole at s = 1. Setting \xi(s) = \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s), one has \xi(s) = \xi(1-s).

Theorem 3: Prime Number Theorem

\pi(x) \sim \frac{x}{\ln x} \text{ as } x \to \infty, equivalently \pi(x) \sim \text{Li}(x). Proved by showing \zeta(s) \neq 0 on \text{Re}(s) = 1.

Worked Examples

For each prime $p$, expand the geometric series: $\frac{1}{1-p^{-s}} = 1 + p^{-s} + p^{-2s} + \cdots$.
$\prod_p \frac{1}{1-p^{-s}} = \prod_p \sum_{k=0}^\infty p^{-ks}$
By unique factorisation every positive integer $n$ appears exactly once as $\prod_i p_i^{k_i}$, contributing $n^{-s}$. So the product equals $\sum_{n=1}^\infty n^{-s} = \zeta(s)$.

Answer: The Euler product $\prod_p (1-p^{-s})^{-1} = \zeta(s)$ follows directly from unique factorisation.

Practice Problems

Difficulty 7/10

Show that $\zeta(2) = \pi^2/6$ is equivalent to the identity $\prod_p \frac{p^2}{p^2-1} = \pi^2/6$.

Difficulty 8/10

Explain why $L(1, \chi) \neq 0$ for a non-principal character $\chi$ mod $q$ implies infinitely many primes $p \equiv a \pmod q$.

Difficulty 9/10

State the Riemann Hypothesis precisely and explain its implication for the prime counting function $\pi(x)$.

Historical Background

Dirichlet introduced his L-functions in 1837 to prove that every arithmetic progression $\{a + nd\}$ with $\gcd(a,d) = 1$ contains infinitely many primes. Riemann's 1859 memoir introduced the analytic continuation and functional equation of $\zeta(s)$ and stated the Riemann Hypothesis. The prime number theorem was proved by Hadamard and de la Vallée Poussin in 1896 using $\zeta(s)$.

1837
Dirichlet introduces L-functions to prove primes in arithmetic progressions
Peter Gustav Lejeune Dirichlet
1859
Riemann's memoir on the zeta function and the Riemann Hypothesis
Bernhard Riemann
1896
Prime number theorem proved via the non-vanishing of zeta on Re(s)=1
Jacques Hadamard, Charles de la Vallée Poussin

Summary

Dirichlet series $\sum a_n n^{-s}$ converge absolutely for $\text{Re}(s)$ large enough.
The Riemann zeta function $\zeta(s) = \sum n^{-s} = \prod_p (1-p^{-s})^{-1}$ extends to all $\mathbb{C}$.
Dirichlet L-functions $L(s,\chi)$ prove primes occur in every coprime arithmetic progression.
Zeros of $\zeta$ control the error in $\pi(x) \approx x/\ln x$.
The Riemann Hypothesis (all non-trivial zeros on $\text{Re}(s)=1/2$) is the central open problem in mathematics.

References

BookDavenport, H. Multiplicative Number Theory. Springer, 2000.
BookTitchmarsh, E. The Theory of the Riemann Zeta-Function. Oxford, 1986.