1.3. L-functions 13 Figure 4. Johann Peter Gustav Lejeune Dirichlet (1805- 1859) and Georg Friedrich Bernhard Riemann (1826-1866). where χ is a given Dirichlet character. For example, one can take χ0 to be the trivial Dirichlet character, i.e., χ0(n) = 1 for all n ≥ 1. Then L(s, χ0) is the Riemann zeta function ζ(s). Dirichlet L-functions also have Euler products: L(s, χ) = ∞ n=1 χ(n) ns = p 1 1 − χ(p)p−s . The idea of the proof of Dirichlet’s theorem generalizes the fol- lowing proof, due to Euler, of the infinitude of the primes. Consider ζ(s) = ∑∞ n=1 1 ns = p 1 1−p−s and suppose there are only finitely many primes. Then the product over all primes is finite, and therefore its value at s = 1 would be finite (a rational number, in fact). However, ζ(1) = ∑∞ n=1 1/n is the harmonic series, which diverges! Therefore, there must be infinitely many prime numbers. Dirichlet adapted this argument by looking instead at a different function: Ψa,N(s) = p≡a mod N 1 ps .
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