Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series

Abstract

Abstract. The present paper contains three main results. The first is asymptotic expansions of Barnes double zeta-functions, and as a corollary, asymptotic expansions of holomorphic Eisenstein series follow. The second is asymptotic expansions of Shintani double zeta-functions, and the third is the analytic continuation of n-variable multiple zeta-functions (or generalized Euler-Zagier sums). The basic technique of proving those results is the method of using the Mellin-Barnes type of integrals.

Zeta-Functions Defined by Two Polynomials

“…For instance, we will prove certain convergent infinite series expansions (Theorem A(iii) and Remark 2). These results show an interesting new feature of our present situation, because in [Mat1,2] we have obtained similar asymptotic expansions for multiple zeta sums but they are not convergent (Remark 3).…”

“…Katsurada [K1,2] discovered that the Mellin-Barnes formula is useful to study the analytic behavior of double zeta sums. The first author [Mat1,2] generalized Katsurada's idea to obtain a proof of meromorphic continuation of Euler-Zagier multiple zeta sums. The method in this paper is a modification of the argument developed in [Mat1,2].…”

On Witten Multiple Zeta-Functions Associated With Semi-Simple Lie Algebras Iv

“…The arrow from G 2 to C 2 in the diagram in [9, Section 5] is horizontal, hence this is the case when the shifting of the path is sufficient. Therefore, our argument here is not so complicated, similar to that in [12,11,13]. Note that such simple shifting argument is not sufficient when one studies analytic properties of zeta-functions of other exceptional algebras.…”

Generalized multiple Dirichlet series and generalized multiple polylogarithms

“…First we prove the following result by using the method introduced by Matsumoto-Tanigawa [14] (see also [11][12][13]). Indeed, this can be regarded as a generalization of Theorem 2 in [14].…”

“…Proof of Theorem 1.1. Using the method introduced in [14, Section 2] (see also [11][12][13]), we give the proof of Theorem 1.1 by induction on r. The case of r = 1 can be directly obtained from Assumptions I and II.…”