On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters.

On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters. (1993)

by Andrew Granville

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Datum	Value	Source
TITLE	On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters.	user correction - Legacy Corrections
AUTHOR NAME	Andrew Granville	user correction - Legacy Corrections
ABSTRACT	: 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=OE(q) for each a 62 H and `(x; q; a) = o(x=OE(q)), otherwise. From here, we deduce that if the second case holds at all, then it holds only for the multiples of some fixed integer q 0 ? 1. Actually, even if the Prime Number Theorem for arithmetic progressions, modulo q, does hold, these methods allow us to deduce the behaviour of a possible `Siegel zero' from Selberg's formula. We also propose a new method for determining explicit upper and lower bounds on `(x; q; a), which uses only elementary number theoretic computations. 1. Introduction. Define `(x) = P px log p, where p only denot...	user correction - Legacy Corrections
YEAR	1993	user correction - Legacy Corrections
CITATIONS	25 found	ParsCit 1.0