1. In [12] and [13] Selberg introduced a trace formula for a compact, locally symmetric space of negative curvature. There is a natural algebra of operators on any such space which commute with the Laplacian. The Selberg trace formula gives the trace of these operators. Selberg also pointed out the importance of deriving such a formula when the symmetric space is assumed only to have finite vol...

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 po...

We study a model sifting problem introduced by Selberg, in which all of the primes have roughly the same size. We show that the Selberg lower bound sieve is asymptotically optimal in this setting, and we use this to give a new lower bound on the sifting limit βκ in terms of the sifting dimension κ. We also show that one can use a rounding procedure to improve on the Selberg lower bound sieve by...

Such identities arise in Rankin-Selberg integral representations of L-functions. For GL2, a naive, direct computation is sufficient. However, for general GLn and other higher-rank groups, direct computation is inadequate. Further, connecting local Rankin-Selberg computations to Schur functions usefully connects these computations to the Shintani-Casselman-Shalika formulas for spherical p-adic W...

In [5] A. Selberg axiomatized properties expected of L-functions and introduced the “Selberg class.” We recall that an element F of the Selberg class S satisfies the following axioms. Axiom 1. In the half-plane σ > 1 the function F (s) is given by an absolutely convergent Dirichlet series ∑∞ n=1 a(n)n −s with a(1) = 1 and a(n) ≪ n for every ǫ > 0. Axiom 2. There is a natural number m such that ...

We derive a generalized Selberg-type zeta function for a smooth de-terministic ow which relates the spectrum of an evolution operator to the periodic orbits of the ow. This relation is a classical analog of the quantum trace formulas and Selberg-type zeta functions.

Abstract In [Sa], it was proved that the Selberg zeta function for SL2(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL2(Z). As applications, we study the Brun-Titc...

Recently, Storm [10] defined the Ihara-Selberg zeta function of a hypergraph, and gave two determinant expressions of it by the Perron-Frobenius operator of a digraph and a deformation of the usual Laplacian of a graph. We present a new determinant expression for the Ihara-Selberg zeta function of a hypergraph, and give a linear algebraic proof of Storm’s Theorem. Furthermore, we generalize the...

We further recall that under Selberg orthonormality conjecture, has unique factorization into primitive functions, the only primitive function with a pole at s = 1 is the Riemann zeta function ζ(s), and Fθ(s) is a primitive function if θ ∈R and if F ∈ are primitive and entire (see [4, Section 4]). We say a primitive function F ∈ is normal if θF = 0. Assuming Selberg orthonormality conjecture, w...

s of papers by Alessandro Zaccagnini Alessandro Zaccagnini Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze, 53/A, Campus Universitario 43124 Parma, Italy. [email protected] April 29, 2013 This file contains full references and abstracts of my papers. Chronologically, these deal with the following topics: • Large gaps between primes in arithme...