Euler products of Selberg zeta functions in the critical strip

Abstract

We extend the region of convergence of Euler products of Selberg zeta functions beyond the boundary R(s) = 1 for congruence subgroups of SL₂(ℤ) if they are associated with nontrivial irreducible unitary representations. The region depends on the size of the lowest eigenvalue of the Laplacian and extends to R(s) ⩾ 3/4 under the Selberg eigenvalue conjecture. The method is based on the ideas of Ramanujan. For any unitary representation, we also establish a relation between the asymptotic behaviour of partial Euler products and the error term in the prime geodesic theorem.

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