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The hybrid mean value of Dedekind sums and two-term exponential sums

Chang Leran, Li Xiaoxue (2016)

Open Mathematics

In this paper, we use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the two-term exponential sums, and give an interesting identity and asymptotic formula for it.

The Lax-Phillips infinitesimal generator and the scattering matrix for automorphic functions

Yoichi Uetake (2007)

Annales Polonici Mathematici

We study the infinitesimal generator of the Lax-Phillips semigroup of the automorphic scattering system defined on the Poincaré upper half-plane for SL₂(ℤ). We show that its spectrum consists only of the poles of the resolvent of the generator, and coincides with the poles of the scattering matrix, counted with multiplicities. Using this we construct an operator whose eigenvalues, counted with algebraic multiplicities (i.e. dimensions of generalized eigenspaces), are precisely the non-trivial zeros...

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