Further arithmetical applications

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

$$\frac{{\phi \left( q \right)}} {2}\left( {1 - \frac{{Log\,q}} {{\log \,N}}} \right)$$

((5.1))

congruence classes modulo q contains a prime ≤ N, so roughly speaking slightly less than ϕ(q)/2 when q is N^ε. 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

$$A\left( {{q_i}} \right) = \left\{ {a \in \mathbb{Z}/{q_i}\mathbb{Z}/\exists p \in p,p \equiv a\left[ {{q_i}} \right]} \right\}$$

.