Zeta function and regularized determinant on a disc and on a cone

Spectral Analysis and Zeta Determinant on the Deformed Spheres

“…for Re(s) > 1+α, and where x and p are real constants subject to the conditions that am+n+x > 0 and (am + n + x) 2 + p > 0 for all n and m, and α is a non negative integer (the case where α is any real number can be treated by similar methods, but is much more complicate, see [56]). …”

“…Despite the lack of a general method, a lot of results are available in the literature for various particular cases or by means of some kind of approximation. Moreover, quite complete results have been obtained for the geometric case of the metric Laplacian on a Riemannian compact manifold for some classes of simple spaces: spheres [18] [11], projective spaces [54], balls [5], orbifolded spheres [25], compact (and non compact) hyperbolic manifolds [20] [9] [10] or in particular cases: Sturm operators on a line segment [8] [45], cone on a circle [56].…”

Zeta invariants for Dirichlet series

“…and again the canonical product converges; moreover, the exponential factor makes no contribution to the quantities we are going to compute, since it adds terms to R(λ, s) that vanish after integration over λ (see [Spreafico 2005]). Also notice that, since T (λ, s) is defined up to a constant, we can always chose it so that T (0, s) = 0, and we do so.…”

“…The Spectral Decomposition Lemma is a theoretical result that becomes effective when a concrete expansion for t n (λ) is available; this is actually the case in many interesting examples, such as the ones we will deal with in the next sections or the one studied in [Spreafico 2005]. …”

“…The lemma we are going to prove is a general version of a technique introduced in [Spreafico 2005] to treat the zeta function associated to the Laplacian on a cone. Let P(x, y) be a polynomial in two variables with real coefficients.…”