Abstract
In this section, we use the large sieve extension of the Brun-Titchmarsh inequality provided by Theorem 2.1 to detect products of two primes is arithmetic progressions. Let us consider the case of primes in [2, N], of which the prime number theorem tells us there are about N/ Log N. Next select a modulus q. The Brun-Titchmarsh Theorem 2.2 implies that at least
congruence classes modulo q contains a prime ≤ N, so roughly speaking slightly less than ϕ(q)/2 when q is Nε. If this cardinality is > ϕ(q)/3, one could try to use Kneser’s Theorem and derive that all invertible residue classes modulo q contain a product of three primes, but the proof gets stuck: all the primes we detect — to show the cardinality is more than ϕ(q)/3 — could belong to a quadratic subgroup of index 2 … However the following theorem shows that if this is indeed the case for a given modulus q then the number of classes covered modulo some q′ prime to q is much larger: Theorem 5.1. Let N ≥ 2. Set P to be the set of primes in \(\left] {\sqrt N ,N} \right]\), of cardinality P, and let (q i ) i∈ I be a finite set of pairwise coprime moduli, not all more than \(\sqrt N /Log\,N\). Define
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© 2009 Hindustan Book Agency
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Ramaré, O. (2009). Further arithmetical applications. In: Ramana, D.S. (eds) Arithmetical Aspects of the Large Sieve Inequality. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-40-8_6
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DOI: https://doi.org/10.1007/978-93-86279-40-8_6
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-90-6
Online ISBN: 978-93-86279-40-8
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