On Elementary Proofs of the Prime Number Theorem for Arithmetic Progressions, without Characters
نویسنده
چکیده
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 case holds at all, then it holds only for the multiples of some xed 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 possiblèSiegel 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.
منابع مشابه
Prime Numbers in Certain Arithmetic Progressions
We discuss to what extent Euclid’s elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet’s theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod k ) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.
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