vi Contents 2.5. Kloosterman sums 47 2.6. Basic estimates 49 Chapter 3. Automorphic Forms 53 3.1. Introduction 53 3.2. The Eisenstein series 56 3.3. Cusp forms 57 3.4. Fourier expansion of the Eisenstein series 59 Chapter 4. The Spectral Theorem. Discrete Part 63 4.1. The automorphic Laplacian 63 4.2. Invariant integral operators on C(Γ\H) 64 4.3. Spectral resolution of Δ in C(Γ\H) 68 Chapter 5. The Automorphic Green Function 71 5.1. Introduction 71 5.2. The Fourier expansion 72 5.3. An estimate for the automorphic Green function 75 5.4. Evaluation of some integrals 76 Chapter 6. Analytic Continuation of the Eisenstein Series 81 6.1. The Fredholm equation for the Eisenstein series 81 6.2. The analytic continuation of Ea(z, s) 84 6.3. The functional equations 86 6.4. Poles and residues of the Eisenstein series 88 Chapter 7. The Spectral Theorem. Continuous Part 95 7.1. The Eisenstein transform 96 7.2. Bessel’s inequality 98 7.3. Spectral decomposition of E(Γ\H) 101 7.4. Spectral expansion of automorphic kernels 104 Chapter 8. Estimates for the Fourier Coefficients of Maass Forms 107 8.1. Introduction 107 8.2. The Rankin–Selberg L-function 109 8.3. Bounds for linear forms 110 8.4. Spectral mean-value estimates 113
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