Quadratic Forms and Closed Geodesics

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The Riemann Hypothesis ◮ Think of ∞∑ n=0 z n = 1 1 − z . Series: |z| < 1, 1/(1 − z), z ≠ 1. ∞∑ 1 ◮ ζ(s) = , ns R(s) > 1. n=1 Extend to 1/2 ≤ R(s), then to C using functional equation Quadratic Forms and Closed Geodesics Y. Petridis Quadratic Forms Hyperbolic surfaces Closed Geodesics Spectral Theory −1+s Γ((1 − s)/2) ζ(s) = π ζ(1 − s). Γ(s/2) Zeros: ρ, 1 − ρ symmetric around R(s) = 1/2. Riemann Hypothesis R(ρ) = 1/2 ALWAYS
The Riemann Hypothesis ◮ Think of ∞∑ n=0 z n = 1 1 − z . Series: |z| < 1, 1/(1 − z), z ≠ 1. ∞∑ 1 ◮ ζ(s) = , ns R(s) > 1. n=1 Extend to 1/2 ≤ R(s), then to C using functional equation Quadratic Forms and Closed Geodesics Y. Petridis Quadratic Forms Hyperbolic surfaces Closed Geodesics Spectral Theory −1+s Γ((1 − s)/2) ζ(s) = π ζ(1 − s). Γ(s/2) Zeros: ρ, 1 − ρ symmetric around R(s) = 1/2. Riemann Hypothesis R(ρ) = 1/2 ALWAYS
Page 1 and 2: Quadratic Forms and Closed Geodesic
Page 3 and 4: Pell’s Equation Quadratic Forms a
Page 9 and 10: Indefinite Quadratic Forms ◮ Q(x,
Page 15 and 16: Class Numbers h(d) = number of ineq
Page 17 and 18: From Quadratic Forms to Hyperbolic
Page 19 and 20: Linear Fractional Transformations a
Page 21 and 22: Geodesics in Hyperbolic Disc Quadra
Page 23 and 24: Fundamental Domains z ∼ w ⇔ w =
Page 25 and 26: ¡ Tesselations Quadratic Forms and
Page 27 and 28: Pell’s Equation and Lengths of Cl
Page 29 and 30: Distribution Closed geodesics of H/
Page 31: The Laplace Operator Quadratic Form
Page 34 and 35: Contour plots of eigenfunctions of
Page 38 and 39: Relations of RH with spectral theor
Page 40 and 41: Duality between periods and eigenva
Page 42 and 43: Simple trace formula Quadratic Form

Quadratic Forms and Closed Geodesics

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