Dirichlet ’ s theorem — set up
نویسنده
چکیده
1 Dirichlet’s theorem — set up Dirichlet’s famous theorem from 1837 asserts that if k is a positive integer and a is an integer coprime to k, then there are infinitely many primes p with p ≡ a (mod k). In this unit we will prove the stronger assertion that the sum of the reciprocals of the primes p ≡ a (mod k) is divergent. Important in the proof is the use of the orthogonality relations for the Dirichlet characters mod k to pick out the residue class a (mod k). First note that if χ is a character mod k and s > 1, then the Euler product L(s, χ) = ∏
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