Zeta invariants for Dirichlet series

Abstract

We introduce a general method for obtaining the main zeta invariants for a class of double series of Dirichlet type and we apply it to the case of homogeneous quadratic and linear double series.

The analytic torsion of a disc

“…Using classical estimates for the zeros of the Bessel functions, it is possible to prove that the relevant sequences U and S are contained in the class of abstract sequences introduced in [24,26]. This means that we can use the method of [23,25,26], to evaluate the derivative at zero of the associated zeta functions. In particular, we will use the notation and the formula as given in Section 4 of [11], and all the reference of the following Sects.…”

“…Using the absolute BC in (25) we have, for the four types, ∂ x ((α (1) n ) θ s )(l, θ 1 , θ 2 ) = ∂ x (x 1 2 J μ n (λ n x))(l) = 0, (β (1) n ) x (l, θ 1 , θ 2 ) = ∂ x ((β (1) n ) θ s )(l, θ 1 , θ 2 ) = ∂ x (x − 1 2 J μ n (λ n x))(l) = 0, (γ (1) n ) x (l, θ 1 , θ 2 ) = J μ n (lλ n ) = 0, ∂ x (γ (1) n ) θ s (l, θ 1 , θ 2 ) = − 1 4 J μ n (lλ n ) + λ n J μ n (lλ n ) + λ 2 n J μ n (lλ n ) = 0,…”

“…(λ 0 x))(l) = 0, and so we obtain the square of the zeros of 1 2 J μ n (lλ n ) + lλJ μ n (lλ n ) = 0, − 1 2 J μ n (lλ n ) + lλJ μ n (lλ n ) = 0, J μ n (lλ n ) = 0, and −λ Using the relative BC in (25) we have, for the five types, (α (1) n ) θ s (l, θ 1 , θ 2 ) = (x 1 2 J μ n (λ n x))(l) = 0, (β (1) n ) θ s (l, θ 1 , θ 2 ) = J μ n (lλ n ) = 0, ∂ x (x 2 (β (1) n ) x )(l, θ 1 , θ 2 ) = − 1 4 J μ n (lλ n ) + lλ n J μ n (lλ n ) + (lλ n ) 2 J μ n (lλ n ) = 0, …”

Zeta determinants on cones and products

“…This is based on [19][20][21][22]. Let S = {a n } ∞ n=1 be a sequence of non vanishing complex numbers, ordered by increasing modules, with unique point of accumulation at infinite.…”

“…In this note we announce some results on the zeta determinant of the Laplace operator on two classes of (compact) geometries: cones and product spaces. These results are obtained by applying some techniques in zeta determinants and regularized products introduced and developed in a series of recent works [19][20][21][22]. In particular, a detailed account and complete proofs can be found in [22].…”