Spectral Determinants of Arithmetic Surfaces and Critical Zeros of Automorphic Zeta Functions

Abstract

This paper establishes a novel connection between the spectral theory of hyperbolic Laplacians on arithmetic surfaces and the Riemann Hypothesis for automorphic L-functions.

Arithmetic Quantum Chaos and Zeros of L-Functions

This research investigates the deep connections between the spectral geometry of hyperbolic surfaces and the distribution of zeros of automorphic zeta functions. The paper builds upon the foundational framework established in arXiv:2602.17348v1, which introduced novel techniques for analyzing quantum ergodicity on non-compact arithmetic surfaces. Here, the authors develop a theory of spectral determinants Δ_Γ(s) attached to Fuchsian groups Γ, demonstrating that these determinants serve as generating functions for the zeros of zeta functions ζ_Γ(s) through their pole structure.

The Explicit Formula and Geodesic Sums

A central result is the derivation of an explicit formula that establishes a duality between the spectrum of the hyperbolic Laplacian and the length spectrum of closed geodesics. Specifically, the authors prove that sums over zeros ρ of ζ_Γ(s) can be expressed as sums over the primitive closed geodesics {γ} on the surface Γ\ℍ, weighted by factors involving the hyperbolic sine of the geodesic length. This relationship takes the form:

• Σ_ρ h(ρ) = Σ_{ {γ} } (ℓ(γ) / (e^{ℓ(γ)/2} - e^{-ℓ(γ)/2})) · ĥ(ℓ(γ)) + [identity contribution]

where h is a suitable test function and ĥ denotes its Fourier transform. This formula reveals that the zeros of the zeta function are in direct correspondence with the quantum energies of the Laplacian, while the geodesic lengths provide the classical periodic orbit contributions.

A Spectral Criterion for the Critical Line

The paper's most significant contribution toward the Riemann Hypothesis is the proof that the location of zeros on the critical line Re(s) = 1/2 is equivalent to the positivity of the spectral determinant. The authors establish that ζ_Γ(s) satisfies the generalized Riemann Hypothesis if and only if Re(log Δ_Γ(1/2 + it)) > 0 for all real t. This transforms the non-vanishing problem into a variational question about the real part of a spectral quantity, offering a new analytic approach to controlling the off-critical behavior of zeta functions.

Computational Verification

The theoretical framework is supported by extensive computational evidence obtained through Wolfram Language implementations of the Selberg trace formula. The authors compute the spectral determinants for congruence subgroups Γ_0(N) with levels N ≤ 100, verifying the positivity condition numerically and confirming that the zeros of the associated zeta functions align with the poles of the determinants as predicted by the theory. These calculations provide empirical support for the conjecture that arithmetic groups possess the spectral positivity required for the Riemann Hypothesis to hold.