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# On Elementary Proofs of the Prime Number Theorem for Arithmetic Progressions, without Characters

@inproceedings{Granville1993OnEP, title={On Elementary Proofs of the Prime Number Theorem for Arithmetic Progressions, without Characters}, author={Andrew Granville}, year={1993} }

- Published 1993

We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that (x; q; a) 2x==(q) for each a 6 2 H and (x; q; a) = o(x==(q)), otherwise. From here, we deduce that if the second… CONTINUE READING