Dirichlet's theorem on primes in arithmetic progressions says that for any sequence s_N := aN + b, with a,b in Z+ with GCD(a,b) = 1, there exists an infinite number of primes s_N. I'm reading detailed versions of his proof, circa 1837. I'm deeply impressed by the complexity of this proof, making use of Fourier analysis on finite abelian groups as well as the theory of (analytic) functions. The various modern versions I'm reading seem to agree almost exactly on the reasoning involved, so I presume that there isn't much to modernize. Question: Is it really true that there isn't an "elementary" proof that avoids analytic functions? It appears that at least such do exist for a number of specific arithmetic progressions, like 6N+1 or 6N-1. Somewhat related question: Given a finite abelian group G, its group G^ of "characters" is the group of all homomorphisms Hom(G,R/Z) from G to the circle group R/Z (the group operation is pointwise multiplication in R/Z). It's easy to see that |G^| = |G|, but to see that these two groups are isomorphic seems to be proved using the classification theorem for finite abelian groups. Is there a proof that G^ is isomorphic to G that doesn't rely on this classification theorem? --Dan ________________________________________________________________________________________ It goes without saying that .