Zeta invariants for sequences of spectral type, special functions and the Lerch formula

Abstract

We discuss the spectral properties of a class of sequences of what we call 'spectral' type. We introduce an effective method to calculate the zeta invariants for this type of sequence. Such invariants are given in terms of some new and old special functions, and we consider a number of examples in which we study the properties of these special functions.

Spectral Analysis and Zeta Determinant on the Deformed Spheres

“…The zeta functions of this type (with integer coefficients) appear when dealing with the zeta functions of a narrow ideal class for a real quadratic field as shown by Zagier in [66] and [67], where he also computes the values at non positive integers (see also [51] [28] [14] [15], and in particular [58] for the derivative). Beside, we can overcome this difficulty in the case under study first by reducing the multi dimensional zeta functions to a sum of 2 dimensional linear and quadratic zeta functions, and then studying the quadratic one by means of a general method introduced in [59] in order to deal with non homogeneous zeta functions. Note that, for particular values of the deformation parameter, the zeta function can be reduced to a sum of zeta functions of Barnes type [3], and this allows a direct computation of the main zeta invariants [25] [26].…”

“…By the following lemma (see [60] or [59]), we can reduce ζ(s, ∆ S N +1 k ) to a sum of simple and double zeta functions.…”

Zeta invariants for Dirichlet series

“…The t > 1 part defines a regular function of s near s = 0, while in the t < 1 part we must change the contour of the λ integral (here C is a circle around the origin of ray ) and then we can rescale λ by t and use the expansion above to obtain (see [Spreafico 2006])…”

“…were first introduced by Barnes in his work on multiple gamma functions (see [Cassou-Noguès 1990] for a good review), where he gives formulas for the values at negative integers and relations with the multiple gamma functions that extend the classical Lerch formula (see also [Spreafico 2006]). Notice also the work of Actor [1990], which gives a formula for ζ (s; a) as a power series whose coefficients are (infinite) sums of Riemann zeta functions and elementary functions and the related works of Matsumoto [1998], where asymptotic expansions are given for nonhomogeneous linear series ζ 1 (s; a, 1, q) = ∞ n,k=1 (an + k + q) −s , for large a, and observe that a formula for ζ 1 (0; a, 1) was given in [Spreafico 2004], using the Plana Theorem.…”

“…The general plan is as follows. We know how to deal with a simple series [Spreafico 2006], say the sum over k, and we can rearrange the sums so as to decompose the double series as a sum over n of sums over k. The problem in this decomposition is that we have to regularize also the external sum in n. For that purpose, it is enough to collect an opportune power of n. This will give a suitable analytic representation and will keep track of all contributions in the regularization process.…”