Xxxxxxxxxx Conjecture). For every A > 0 and 0 < θ < 1 we have that q≤xθ (a,q)=1 n≤x φ(q) logA x Σ max Σ Λ(n) − x ≪ x . n≡a(q) In 2013, Xxxxx [44] in breakthrough work established unconditionally that there exist infinitely many bounded gaps between the primes. In particular, Xxxxx showed that − × lim inf(pn+1 pn) < 7 107. n→∞ Later in 2013, Xxxxxxx [30] introduced a new idea which substantially sim- plified the proof of this result and established the improved bound 600 for (1.0.5) unconditionally. Under the Xxxxxxx-Xxxxxxxxxx conjecture, Xxxxxxx obtains the bound 12. The Polymath8b project [36] subsequently improved the unconditional bound to 246, and under the assumption of the generalised Xxxxxxx-Xxxxxxxxxx conjecture obtained (1.0.5) with 6. The twin prime con- jecture would amount to proving (1.0.5) with the bound 2. Xxxxxxxx, Xxxxxx, Xxxxx and Xxxxxxxx [11] proved an almost prime xxx- xxxxx of (1.0.5) which holds unconditionally; if q1 < q2 < · · · denotes the sequence of products of exactly two distinct primes, then − ≤ lim inf(qn+1 qn) 6. n→∞ These results on bounded gaps between primes and almost primes are proved using sieve theory and the arguments do not establish an asymptotic formula. However, sieve methods can be used to obtain an upper bound for the correlation (1.0.4), namely Σ 1 X X<n≤2X S(h) 1P(n)1P(n + h) ≪ (log X)2 . (1.0.6) Xxxx’x theorem gives that p + 2 = q such that p is prime and q is either a prime or a product of two primes holds for infinitely many primes p. Debouzy [5] proved under the Xxxxxxx-Xxxxxxxxxx conjecture that given any 0 ≤ β < γ there exists X0 such that for all X ≥ X0 we have that Σ Λ(n)Λ(n+2)+ 1 Σ Λ(n+2) Σ Λ(d1)Λ(d2) = 2Π X(1+o(1)). n≤X γ − β n≤X d1d2=n nβ≤d1≤nγ This result is proved using an improvement of the Bombieri asymptotic sieve. More generally, Bombieri [2] had previously considered pairs Pk and Pk + 2 = p with p prime and Pk an almost prime with at most k factors. More precisely, defining Λk(n) := (µ ∗ logk)(n) to be the generalised von Mangoldt function where ∗ denotes Dirichlet convolution, Bombieri proved that if k ≥ 1 is an integer and x ≥ x0(k) we have Σ Λk(n)Λ(n + 2) = 2Π2X(log X)k−1(k + O(k4/32−k/3)) n≤X and, assuming the Xxxxxxx-Xxxxxxxxxx conjecture, for k ≥ 2 we have the asymptotic Σ Λk(n)Λ(n + 2) ∼ 2Π2kX(log X)k−1. (1.0.7) n≤X
Xxxxxxxxxx Conjecture). ( )∞ 𝑝 In this subsection we consider the following question that constitutes a further strengthening of µVan𝐸 ,Z 1 which, for convenience, we will sometimes refer to as Xxxxxxxxx’x conjecture for 𝑈 -ray class group modulo Σ: Question 2.8.19 (Gr𝐸,𝑈 ,Σ). Suppose that 𝐸 is a totally real field and that 𝐸∞ is the cyclotomic Z𝑝 - extension. Then lim ←–𝑛 eN ClΣ 𝑈 (𝐸𝑛) (𝐸𝑛)𝑝 is a pseudo-null Λ-module. Remark 2.8.20.
