Zeta Determinant for Double Sequences of Spectral Type and a Generalization of the Kronecker First Limit Formula

The Kronecker first limit formula in the number field K is an expression of the value of the constant term in the Laurent expansion at s = 1 of the partial Dedekind zeta function ζA(s), associated to an ideal class A of K (see Section 4.1 for details), and has deep applications in number theory. When K is the rational number field or an imaginary quadratic field such formulas are classical and well understood. The case of a real quadratic field is much harder. It was originally tackled by Hecke, and has been investigated in the fundamental works of Shitani [23] [24] and Zagier [33] [34]. As shown in [33], the relevant zeta function is the zeta function associated to a quadratic form, namely a double zeta function of Dirichlet type like ζ2(s; a, b, c) = ∑∞ m,n=1(am 2+2cmn+bn2)−s, where a, b and c are positive real numbers (see Section 3.1 for details). Beside multiple zeta functions have been deeply studied, the analysis is usually based on the Fourier expansion of the theta function that applies when sums over the whole integer ring appear (see for example [6]), namely when the zeta functions reduce to zeta functions of Eisenstein type. When we are concerned with Dirichlet sums, namely with sums over the natural integers, we loose a great amount of symmetries and the Poisson summation formula does not apply. As a result it is highly non trivial to obtain a generalization of the Kronecker formula for Dirichlet series. However, in the recent work on zeta regularized products, the Kronecker first limit formula received a renewed interest, and various generalizations and different approaches have been investigated [32] [18] [17] [16] [28]. This interest is also motivated because the evaluation of the constant term in the expansion at s = 1 of the relevant zeta function is equivalent, in presence of a functional equation, to the study of the derivative of the zeta function at s = 0. The latter problem became of fundamental importance in various areas of mathematics and theoretical physics, since the value of the derivative of the zeta function at zero was used by Ray and Singer [22] in order to define the functional determinant of a linear operator. In this note we pursuit the attempt of tackling the problem under a unified approach, namely we present a general method for evaluating the first terms in the asymptotic expansion of a class of abstract double zeta functions. The idea of a unified approach is not new, and has been developed with success by various authors (see Section 2 for more details and references on this subject). In particular, we introduced and investigated the zeta functions of a class of abstract sequences of