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On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters.
, 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 s ..."
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: 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...
HYPOTHESIS H AND AN IMPOSSIBILITY THEOREM OF RAM MURTY
, 2010
"... Dirichlet’s 1837 theorem that every coprime arithmetic progression a mod m contains infinitely many primes is often alluded to in elementary number theory courses but usually proved only in special cases (e.g., when m=3 or m=4), where the proofs parallel Euclid’s argument for the existence of infin ..."
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Dirichlet’s 1837 theorem that every coprime arithmetic progression a mod m contains infinitely many primes is often alluded to in elementary number theory courses but usually proved only in special cases (e.g., when m=3 or m=4), where the proofs parallel Euclid’s argument for the existence of infinitely many primes. It is natural to wonder whether Dirichlet’s theorem in its entirety can be proved by such “Euclidean ” arguments. In 1912, Schur showed that one can construct an argument of this type for every progression a mod m satisfying a 2 ≡ 1 (mod m), and in 1988 Murty showed that these are the only progressions for which such an argument can be given. Murty’s proof uses some deep results from algebraic number theory (in particular the Chebotarev density theorem). Here we give a heuristic explanation for this result by showing how it follows from Bunyakovsky’s conjecture on prime values of polynomials. We also propose a widening of Murty’s definition of a Euclidean proof. With this definition, it appears difficult to classify the progressions for which such a proof exists. However, assuming Schinzel’s Hypothesis H, we show that again such a proof exists only when a 2 ≡ 1 (mod m).