Complete asymptotic expansions associated with Epstein zeta-functions

Let Q(u,v)=|u+vz|² be a positive-definite quadratic form with a complex parameter z=x+iy in the upper-half plane. The Epstein zeta-function attached to Q is initially defined by ζ Z2 (s;z)=∑ m,n=-Q(m,n)^{-s} for Re∈s>1, where the term with m=n=0 is to be omitted. We deduce complete asymptotic expansions of ζ Z {2}}(s;x+iyas y→+∞ (Theorem 1 in Sect. 2), and of its weighted mean value (with respect to y) in the form of a Laplace-Mellin transform of ζ Z {2}}(s;z) (Theorem 2 in Sect. 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of ζ _{\mathbb {Z}^{2}}(s;z) over the whole s-plane is prepared by means of Mellin-Barnes integral transformations (Proposition 1 in Sect. 3). This procedure, differs slightly from other previously known methods of the analytic continuation, gives a new alternative proof of the Fourier expansion of ζ {Z} {2}}(s;z) (Proposition 2 in Sect. 3). The use of Mellin-Barnes type of integral formulae is crucial in all aspects of the proofs; several transformation properties of hypergeometric functions are especially applied with manipulation of these integrals.