Dirichlet’s L-functions and Siegel’s theorem

Abstract

A character of a finite abelian group G is a complex-valued function, not identically zero, defined on the group, such that if A ∈ G, B ∈ G, then χ(A B) = χ(A)χ(G) where A B is the group-composite, of A and B. If E denotes the unit element of G, and A_-1 the group inverse of A ∈ G, we assume as known the following properties of characters:

• (1) if G is of order h, χ(A) is an hth root of unity, and the character defined by the property χ₁(A) = 1 for every A ∈ G, is called the principal character of G;

• (2) an abelian group of order h has exactly h characters;

• (3) χ(E)= 1, for every character χ of G, and given any A ∈ G, A ≠ E, there exists a character χ, such that χ(A)≠ 1;

• (4) the characters of G again form a finite, multiplicative, abelian group of which the principal character is the unit element;

• (5)
  $$\sum\limits_A {\chi (A) = } \left\{ {\begin{array}{*{20}{c}} {h,\quad if\quad \chi = {\chi _1},} \\ {o,\quad if\quad \chi \ne {\chi _1};\quad } \\ \end{array} } \right.\,\sum\limits_A {\chi (A) = } \left\{ {\begin{array}{*{20}{c}} {h,\quad if\quad A = E,} \\ {o,\quad if\quad A \ne E.\quad } \\ \end{array} } \right.$$