COMPLETE ASYMPTOTIC EXPANSIONS ASSOCIATED WITH EPSTEIN ZETA-FUNCTIONS II (SUMMARIZED VERSION)

Abstract

Abstract. The present article announces the results in the forthcoming paper [Ka13]. Let Q(u, v) = ju+vzj2 be a positive-definite quadratic form with a complex parameter z = x + iy in the upper half-plane. The Epstein zeta-function ζZ2(s; z) attached to Q is initially defined by (1.3) below. We have established in the preceding paper [Ka10] complete asymptotic expansions of ζZ2(s;x+ iy) as y! +1, and those of its weighted mean value (with respect to y) in the form of a Laplace-Mellin transform (1.4). The forthcoming paper [Ka13] proceeds further with our previous study to show that similar asymptotic series still exist for a more general Epstein zeta-function ψZ2(s; a, b;µ, ν; z) defined by (1.2) below (see Theorem 1), and also for the Riemann-Liouville transform (1.5) of ζZ2(s; z) (see Theorem 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of ψZ2(s; a, b;µ, ν; z) over the whole s-plane is prepared by means of Mellin-Barnes integral transforms (see Proposition 1 in Section 3). This pro-cedure, differs slightly from other previously known methods of analytic continuation, provides the meromorphic continuation of ψZ2(s; a, b;µ, ν; z) in the form of a double in-finite series (see (2.9) with (3.8) and (3.9)), which is most appropriate for deriving the asymptotic expansions in question. The use of Mellin-Barnes type integrals is crucial in all aspects of the proofs; several transformation and connection formulae for hypergeo-metric functions are especially applied with manipulation of these integrals. 1

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