We give higher-power generalizations of the classical Lerch formula for the gamma function.

Automorphic distributions are distributions on ${ℝ}^d$, invariant under the linear action of the group $SL(d,\mathbb{Z})$. Combs are characterized by the additional requirement of being measures supported in ${\mathbb{Z}}^d$: their decomposition into homogeneous components involves the family ${\left({𝔈}_{i\lambda }^d\right)}_{\lambda \in ℝ}$, of Eisenstein distributions, and the coefficients of the decomposition are given as Dirichlet series $𝒟\left(s\right)$. Functional equations of the usual (Hecke) kind relative to $𝒟\left(s\right)$ turn out to be equivalent to the invariance of the comb under some modification...

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray–Singer torsion on any $3$-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-like trace formulae, that hold also in variable...

Let $\Gamma \setminus {ℍ}^3$ be a finite-volume quotient of the upper-half space, where $\Gamma \subset \mathrm{SL}(2,ℂ)$ is a discrete subgroup. To a finite dimensional unitary representation $\chi$ of $\Gamma$ one associates the Selberg zeta function $Z(s;\Gamma ;\chi )$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tilde{\Gamma }$ is a finite index group extension of $\Gamma$ in $\mathrm{SL}(2,ℂ)$, and $\pi ={\mathrm{Ind}}_{\Gamma }^{\tilde{\Gamma }}\chi$ is the induced representation, then $Z(s;\Gamma ;\chi )=Z(s;\tilde{\Gamma };\pi )$. In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $\phi (s;\Gamma ;\chi )=\phi (s;\tilde{\Gamma };\pi )$, for an appropriate...