Dirichletmerkkejä

Dirichlet characters are a fundamental concept in analytic number theory. They are functions defined on the set of integers that are periodic and multiplicative. Specifically, a Dirichlet character modulo n is a function $\chi: \mathbb{Z} \to \mathbb{C}$ such that:

1. $\chi(a) = 0$ if $\gcd(a, n) > 1$.

2. $\chi(a) = \chi(b)$ if $a \equiv b \pmod{n}$.

3. $\chi(ab) = \chi(a)\chi(b)$ for all integers a and b (multiplicativity).

4. $\chi(1) = 1$.

The modulus n is a positive integer. The set of all Dirichlet characters modulo n forms a group under multiplication, and this group is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^\times$, the multiplicative group of integers modulo n.

There is always a trivial character modulo n, denoted by $\chi_0$, which is defined as $\chi_0(a) = 1$ if $\gcd(a, n) = 1$ and $\chi_0(a) = 0$ if $\gcd(a, n) > 1$. All other characters are called non-trivial.

Dirichlet characters are crucial for studying the distribution of prime numbers in arithmetic progressions, a topic pioneered by Peter Gustav Lejeune Dirichlet. The Dirichlet L-function associated with a Dirichlet character $\chi$ is defined as $L(s, \chi) = \sum_{k=1}^\infty \frac{\chi(k)}{k^s}$ for $\operatorname{Re}(s) > 1$. This function plays a central role in the proof of Dirichlet's theorem on arithmetic progressions.

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