Spectral Zeta Function of the Laplacian

The spectral zeta ζ_Δ(s) = Σ λₙ⁻ˢ encodes the geometry of a manifold through its eigenvalue spectrum. Explore heat kernel trace and spectral invariants.

40
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0.30
ζ_Δ(s) = Σ λₙ⁻ˢ (Re s > d/2)

Heat kernel: K(t) = Σ e^{−λₙt}
Weyl law: λₙ ~ (2π)² (n/Area)^{2/d}

Spectral determinant: det Δ = exp(−ζ'(0))

Poles of ζ_Δ encode geometric invariants (area, perimeter, curvature) via heat kernel expansion.
Computing...