Dirichlet Theorem

Dirichlet Character

Consider the multiplicative group of

q

for

q \in ℤ

. A homomorphism between the 2 groups

χ : {(ℤ / q ℤ)}^{\times} \to ℂ^{\times}

is called a Dirichlet Character modulus q if

\forall a, b \in ℤ

$χ (a b) = χ (a) χ (b)$ , i.e. $χ$ is completely multiplicative (also directly by property of homomorphism)
$χ (a) = {\begin{matrix} 0 & g c d (a, q) > 1 \\ χ (a mod q) & g c d (a, q) = 1 \end{matrix}$
$χ (a + q) = χ (a)$ , i.e. $χ$ is periodic with period q

For every $q \in ℤ$ , we defined a trivial character as $χ_{0} (g) = 1, \forall g \in {(ℤ / q ℤ)}^{\times}$ . This is also called principle character mod q.
Since by the fact that $| {(ℤ / q ℤ)}^{\times} | = ϕ (q)$ and Euler’s theorem, $\forall a \in {(ℤ / q ℤ)}^{\times}$ , $χ (a^{ϕ (q)}) = χ {(a)}^{ϕ (q)} = 1$ . Thus $\forall a \in {(ℤ / q ℤ)}^{\times}$ , $χ (a)$ is a $ϕ (q)$ th roots of unity.

Dirichlet character mod 4

Table of Dirichlet character mod 4

Even/Odd Character

A Dirichlet character is called even if

χ (- 1) = 1

and odd if

χ (- 1) = - 1

It is clear that the trivial character is even.

If p is prime, then

{(ℤ / p ℤ)}^{\times}

is cyclic.

Let p be a prime.

Remark: the generator(s) of ${(ℤ / p ℤ)}^{\times}$ is called primitive roots modulus p.

χ \neq χ_{0}

, then

\sum_{a (m o d q)} χ (a) = 0.

1+2=1

\sum_{χ (m o d q)} χ (n) = {\begin{matrix} ϕ (q) & i f n \equiv 1 (m o d q) \\ 0 & o t h e r w i s e \end{matrix}

1+2=1

Dirichlet Theorem

For any natural number q, and a reduced residue class a (mod q), there are infinitely many primes p ≡ a (mod q). Namely,

\lim_{s \to 1^{+}} \sum_{p \equiv a (m o d q)} \frac{1}{p^{s}} = \infty

Dirichlet L-function

For

s \in ℂ

with

ℜ (s) > 1

, the Dirichlet L-function attached to character

χ

is defined as

L (s, χ) = \sum_{n = 1}^{\infty} \frac{χ (n)}{n^{s}}

Remark:

The Riemann zeta‐function is a special case of L-function where the character attached is the trivial character, namely $L (s, χ_{0}) = ζ (s) = \sum_{n = 1}^{\infty} \frac{1}{n^{s}}$
Dirichlet L-function is absolutely convergent for $ℜ (s) > 1$ .
Just as the Riemann zeta‐function, the Dirichlet L-function can also be analytically continued to a meromorphic function with simple pole at $s = 1$ and statisfied the functional equation.

Euler product of L-function

For

ℜ (s) > 1

L (s, χ) = \prod_{p p r i m e} {(1 - χ (p) p^{- s})}^{- 1}