Let Q(u, v) = |u + vz|<sup>2</sup> be a positive-definite quadratic form with a complex parameter z = x +iy in the upper half-plane. The Epstein zeta-function ζZ<sup>2</sup> (s; z) attached to Q is initially defined by (1.3) below.We have established in the preceding paper Katsurada (Ramanujan J 14:249–275, 2007) complete asymptotic expansions of ζZ<sup>2</sup> (s; x + iy) as y → +∞, and those of its weighted mean value (with respect to y) in the form of a Laplace–Mellin transform (1.4). The present paper proceeds further with our previous study to show that similar asymptotic series still exist for a more general Epstein zeta-function ψZ<sup>2</sup> (s; a, b; μ, ν; z) defined by (1.2) below (Theorem 1), and also for the Riemann–Liouville transform (1.5) of ζZ<sup>2</sup> (s; z) (Theorem 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of ψZ<sup>2</sup> (s; a, b; μ, ν; z) over the whole s-plane is prepared by means of Mellin–Barnes integral transforms (Proposition 1 in Sect. 3). This procedure differs slightly from other previously known methods of analytic continuation, and provides the meromorphic continuation of ψZ<sup>2</sup> (s; a, b; μ, ν; z) in the form of a double infinite series [see (2.9) and (3.9) with (3.8)], which is most appropriate for deriving the asymptotic expansions in question. The use of Mellin–Barnes type integrals such as in (3.3) is crucial in all aspects of the proofs; several transformation and connection formulae for hypergeometric functions are especially applied with manipulation of these integrals.
ASJC Scopus subject areas
- Algebra and Number Theory