An inequality for the modified Selberg zeta-function

The Ramanujan Journal - We consider the absolute values of the modified Selberg zeta-function at places symmetric with respect to the critical line. We prove an inequality for the modified Selberg...     

The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations

For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification points to the zeta-function. In fact, if is the ring of Eisenstein integers, then the Selberg zeta-function of contains ramification points and is the sixth-root of a meromorphic function.     

Note on zeros of the derivative of the Selberg zeta-function

Wenzhi Luo studied the distribution of nontrivial zeros of the derivatives of Selberg zeta-functions on cocompact hyperbolic surfaces, and obtained an asymptotic formula for zero density with bounded height. Then he related the distribution of zeros to the multiplicities of Laplacian eigenvalues. Using better bounds for the growth of the Selberg zeta-functions we improve some of the above results. This work was partially supported by a Grant from the Lithuanian Foundation of Studies and Science.     

The Selberg trace formula and Selberg zeta-function for cocompact discrete subgroups of SO+(1,n)

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 3, pp. 55–64, July–September, 1992.     

Analogues of the Artin factorization formula for the automorphic scattering matrix and Selberg zeta-function associated to a Kleinian group

For Kleinian groups acting on a hyperbolic three-space, we prove factorization formulas for both the Selberg zeta-function and the automorphic scattering matrix. We extend results of Venkov and Zograf from Fuchsian groups to Kleinian groups, and we give a proof that is simple and extendable to more general groups.

     

The reciprocity relation of the Selberg function

The reciprocity relation of the Beta function is reformulated, and that of the Selberg function is presented.     

The Atkinson formula for the periodic zeta-function

We give an Atkinson-type formula for the periodic zeta-function ζ_λ with rational parameter λ. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 504–516, October–December, 2007.     

The Selberg Z-Function. Local Approach

A local variant of the Selberg Z-function is applied to construct its analytic extension and to obtain some estimates in the critical band. Bibliography: 6 titles     

Nielsen zeta-function

In this paper we introduce a new zeta-function in the theory of dynamical systems. We find a sharp bound for the radius of convergence of the Nielsen zeta-function in terms of the topological entropy of the map. It follows from this that the Nielsen zeta-function has a positive radius of convergence. We prove that for an orientation-preserving homeomorphism of a compact surface the Nielsen zeta-function is either a rational function or the radical of a rational function. We calculate the Nielsen zeta-function for maps of circles, spheres, tori, protective spaces, for expanding maps of an orientable smooth compact manifold, for a homotopy periodic map of a connected compact polyhedron having no locally separating point.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 156–161, 1985.     

The Mellin transform of the square of Riemann's zeta-function

The meromorphic continuation of the function. can be established by a method of Y. Motohashi. We show this by an alternative argument revealing the close afinity of this function with 2 (s).     

Growth of the Lerch zeta-function

It is believed that the Lindelöf hypothesis is also true for the Lerch zeta-function. Here we present results supporting this conjecture. We first consider the growth of the Lerch zeta-function assuming the generalized Lindelöf hypothesis for Dirichlet L-functions. We next prove that Huxleys exponent 32/205 in the Lindelöf hypothesis for the Riemann zeta-function holds also for the Lerch zeta-function.__________Partially supported by a grant from the Lithuanian State Science and Studies Foundation.Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 45–56, January–March, 2005.     

The Selberg Z-Function and the Lindelöf Conjecture

A local version of the Selberg Z-function is used in order to extend it analytically and to prove some estimates in the critical band. Bibliography: 10 titles.     

The zero-free region of the derivative of Selberg zeta functions

In this article, we study the zero-free region of the derivative of Selberg zeta functions associated with compact Riemann surfaces and three-dimensional compact hyperbolic spaces. We obtain the zero-free region with respect to the left of the critical line of each Selberg zeta function. This is an improvement of Wenzhi Luo’s zero-free region theorem for compact Riemann surfaces.     

The zero-free region of the derivative of Selberg zeta functions

In this article, we study the zero-free region of the derivative of Selberg zeta functions associated with compact Riemann surfaces and three-dimensional compact hyperbolic spaces. We obtain the zero-free region with respect to the left of the critical line of each Selberg zeta function. This is an improvement of Wenzhi Luo’s zero-free region theorem for compact Riemann surfaces.