Expanding the Exponential Sum Sample Clauses

Expanding the Exponential Sum. ‌ First, we rewrite the integral over the major arcs by expanding the exponen- tial sum S(α) in terms of Dirichlet characters. We first define the following. Definition 2.3.8. Let α = a/q + β satisfy (2.3.4), P be defined as in Defi- nition 2.3.1 and M as in (2.3.4). We define µ(q) a(α) := φ(q) P<pΣ≤P 1+δ 1 Σ p X<n≤2X e(βn), b(α) := φ(q) Σ τ (χ)χ(a) Σ ,ϖ2(n)χ(n) − δχ Σ 1  e(βn), p ∫ A2(X) := ∫ M B2(X) := M χ(q) 2 |a(α)| dα, |b(α)| dα, X<n≤2X P<p≤P 1+δ where τ (χ) denotes the Gauss sum as defined in (2.2.1), and δχ = 1 when χ = χ0 and is zero otherwise. We will now find the following expression for the integral over the major arcs, once we have expanded the exponential sum: Lemma 2.3.9. Let M be defined as in (2.3.4) and a(α), A2(X), B2(X) be as in Definition 2.3.8. We have that ∫ |S(α)|2e(−hα)dα = ∫ |a(α)|2e(−hα)dα + O A(X)B(X) + B2(X) . M M Proof. Let α ∈ M, so that α = a + β with q ≤ Q0, (a, q) = 1 and |β| ≤ 1 . Then q S(α) = Σ qQ 2 q ϖ (n)e an e(βn). X<n≤2X By Definition 2.3.2, we have that n = p1p2 with P < p1 ≤ P 1+δ. As we have P > Q0, we must have that (p1, q) = (p2, q) = 1 and therefore that (n, q) = 1. We can now rewrite our expression for S(α) by applying the identity q φ(q)
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