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Aachen algebraic number approximates a finite becomes chapter character group character mod Characters in Number complex numbers congruence continuous functions declaration modulus deduced defined different from zero Dirichlet characters Dirichlet series Dirichlet series f(s Dirichlet's theorem divisor function equation exponent finite Abelian group finite limit finite number Gauss and Jacobi group with multiplication Hasse Hal homomorphisms infinite integers inverse elements Jacobi sums Krl9 L-series left side Lemma Liouville function natural number number of primes number theoretic functions odd numbers prime residue class prime to 2D proof of Dirichlet's quadratic character quadratic residue r-tuples rational numbers real s>l relatively prime residue class character residue class group residue classes mod roots of unity sequence series L3 Shaker Verlag Springer-Verlag subgroup subseries sum is taken surjective theory of numbers trivial character Uwe Kraeft values whence follows Xr are nontrivial Zahlentheorie zeta-function