Common use of Proof sketch Clause in Contracts

Proof sketch. ‌ We now discuss the main ideas of the proof of Theorem 1.0.2. We apply the ▇▇▇▇▇-▇▇▇▇▇▇▇▇▇▇ circle method (see, for example, [42]), first expressing the correlation Σ 1E′ (n)1E′ (n + h) in terms of the integral ∫ 1 2 2 X<n≤2X Σ 1E′ (n)e(nα) 0 X<n≤2X e(−hα)dα. (2.1.1) We need to understand which points on the unit circle contribute the main term. Dirichlet’s approximation theorem states that for each Q ≥ 1 there exists a/q ∈ Q with (a, q) = 1, 1 ≤ q ≤ Q and |α − a/q| ≤ 1/(qQ). So, we first aim to understand the behaviour of the exponential sum appearing in (2.1.1) at a rational point a/q with (a, q) = 1 on the unit circle. We have that X<Σn≤2X 1E′ (n)e an = Σb=1 Σ Σ X<n≤2X n≡b mod q e ab Σ 1E′ (n) Σ 1. b=1 P<p1≤P 1+δ X <p2≤ 2X p1 p1