Contents vii 8.5. The case of congruence groups 116 Chapter 9. Spectral Theory of Kloosterman Sums 121 9.1. Introduction 121 9.2. Analytic continuation of Zs(m, n) 122 9.3. Bruggeman–Kuznetsov formula 125 9.4. Kloosterman sums formula 128 9.5. Petersson’s formulas 131 Chapter 10. The Trace Formula 135 10.1. Introduction 135 10.2. Computing the spectral trace 139 10.3. Computing the trace for parabolic classes 142 10.4. Computing the trace for the identity motion 145 10.5. Computing the trace for hyperbolic classes 146 10.6. Computing the trace for elliptic classes 147 10.7. Trace formulas 150 10.8. The Selberg zeta-function 152 10.9. Asymptotic law for the length of closed geodesics 154 Chapter 11. The Distribution of Eigenvalues 157 11.1. Weyl’s law 157 11.2. The residual spectrum and the scattering matrix 162 11.3. Small eigenvalues 164 11.4. Density theorems 168 Chapter 12. Hyperbolic Lattice-Point Problems 171 Chapter 13. Spectral Bounds for Cusp Forms 177 13.1. Introduction 177 13.2. Standard bounds 178 13.3. Applying the Hecke operator 180 13.4. Constructing an amplifier 181 13.5. The ergodicity conjecture 183 Appendix A. Classical Analysis 185 A.1. Self-adjoint operators 185
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