Four dimentional joint moments due to Dirichlet density and their applications in summability of quadruple hypergeometric functions

Using the Exton’s multiple joint moments in four dimensional spaces due to Dirichlet density and a generalization of Bosanquet and Kestelman theorem , we prove some theorems in summability of the series containing quadruple hypergeomtric functions. These theorems generalize some well known generating functions and multiplication theorems involving product of hypergeometric functions of one and more variables. We discuss some other applications and establish several interesting particular cases. Finally, we obtain an approximation formula of the series involving Exton’s quadruple hypergeometric function.

Reductions of one-dimensional tori

Consider a non-split one-dimensional torus defined over a number field [Formula: see text]. For a finitely generated group [Formula: see text] of rational points and for a prime number [Formula: see text], we investigate for how many primes [Formula: see text] of [Formula: see text] the size of the reduction of [Formula: see text] modulo [Formula: see text] is coprime to [Formula: see text]. We provide closed formulas for the corresponding Dirichlet density in terms of finitely many computable parameters. To achieve this, we determine in general which torsion fields and Kummer extensions contain the splitting field.

Automorphy and irreducibility of some l-adic representations

In this paper we prove that a pure, regular, totally odd, polarizable weakly compatible system of $l$-adic representations is potentially automorphic. The innovation is that we make no irreducibility assumption, but we make a purity assumption instead. For compatible systems coming from geometry, purity is often easier to check than irreducibility. We use Katz’s theory of rigid local systems to construct many examples of motives to which our theorem applies. We also show that if $F$ is a CM or totally real field and if ${\it\pi}$ is a polarizable, regular algebraic, cuspidal automorphic representation of $\text{GL}_{n}(\mathbb{A}_{F})$, then for a positive Dirichlet density set of rational primes $l$, the $l$-adic representations $r_{l,\imath }({\it\pi})$ associated to ${\it\pi}$ are irreducible.

An Extension of the Dirichlet Density for Sets of Gaussian Integers

Several measures for the density of sets of integers have been proposed, such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and investigate some of its properties.

Bowen's equidistribution theory and the Dirichlet density theorem

Let φ be an Axiom A flow restricted to a basic set, let g be a C∞ function and let , where λg(τ) is the g length of the closest orbit τ, λ(τ) is the period of τ and h is the topological entropy of φ. We obtain an asymptotic formula for πg which includes the ‘prime number’ theorem for closed orbits. This result generalizes Bowen's theorem on the equidistribution of closed orbits. After establishing an analytic extension result for certain zeta functions the proofs proceed by orthodox number theoretical techniques.