THE RAMANUJAN JOURNAL, Vol. 14, No. 2 (2007), 249-275.

# Complete asymptotic expansions associated with Epstein zeta functions

Department of Mathematics, Keio University, 4-1-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8521, Japan

Received January 5, 2004; Accepted September 24, 2004

Abstract. Let $Q(u, v) = \abs{u + v z}^2$ 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 $\zeta_{\Z^2}(s; z) = \sum_{m,n=-infty}^\infty Q(m, n)^{-s}$ for $\mbox{Re} s > 1$, where the term with $m = n = 0$ is to be omitted. We deduce complete asymptotic expansions of $\zeta_{\Z^2}(s; x + iy)$ as $y \to +\infty$ (Theorem 1 in Sect. 2), and of its weighted mean value (with respect to $y$) in the form of a Laplace-Mellin transform of $\zeta_{\Z^2}(s; z)$ (Theorem 2 in Sect. 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of $\zeta_{\Z^2}(s; z)$ 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 $\zeta_{\Z^2}(s; z)$ (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.

Keywords: Epstein zeta-function, Riemann zeta-function, Laplace-Mellin transform, Mellin-Barnes integral, weighted mean value, asymptotic expansion

2000 Mathematics Classification: Primary 11E45; Secondary 11F11