### A generalization of Lerch’s formula

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

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

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

Automorphic distributions are distributions on ${\mathbb{R}}^{d}$, 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 ${\left({\U0001d508}_{i\lambda}^{d}\right)}_{\lambda \in \mathbb{R}}$, of Eisenstein distributions, and the coefficients of the decomposition are given as Dirichlet series $\mathcal{D}\left(s\right)$. Functional equations of the usual (Hecke) kind relative to $\mathcal{D}\left(s\right)$ 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 $\Gamma \setminus {\mathbb{H}}^{3}$ be a finite-volume quotient of the upper-half space, where $\Gamma \subset \mathrm{SL}(2,\u2102)$ 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 $\mathrm{SL}(2,\u2102)$, 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...

We investigate zeta regularized products of rational functions. As an application, we obtain the asymptotic expansion of the Euler Gamma function associated with a rational function.

We prove that the error term ${\sum}_{n\le x}\Lambda \left(n\right)/n-logx+\gamma $ differs from (ψ(x)-x)/x by a well controlled function. We deduce very precise numerical results from the formula obtained.

We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic $L$-functions of ${\mathrm{GL}}_{d}$ over a general number field. This is a generalization of the result of Deninger about the regularized determinant for zeros of the Riemann zeta function.

For a cocompact group $\Gamma $ of ${\text{SL}}_{2}\left(\mathbb{R}\right)$ we fix a real non-zero harmonic $1$-form $\alpha $. We study the asymptotics of the hyperbolic lattice-counting problem for $\Gamma $ under restrictions imposed by the modular symbols $\u2329\gamma ,\alpha \u232a$. We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.