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A Note on Dirichlet's Theorem

A well-known theorem of Dirichlet asserts that for any pair of positive integers k and n which are relatively prime, there exists an infinity of primes p = k (mod n). In 1949, Selberg [5] gave a rather complicated non-analytic proof of this result. Although a simple-minded non-analytic proof of this result is highly desirable, no such proof seems to exist except for special cases. Such a proof was given in 1903 by Birkhoff and Vandiver [1] for the special case k = 1 ; variations of this proof have been given by Rotkiewicz [4] and Estermann [2]. The combinatorial nature of these proofs (for k = 1) unfortunately distract from the underlying simplicity of the idea of the proof. The following is an arithmetic interpretation of Birkhoff and Vandiver's argument. To show the existence of infinitely many primes p = l mod n, it suffices to show there exists infinitely many primes p for which the congruence