caa a22 4500 463178076 CHVBK 20210128105055.0 cr unu---uuuuu 170326e20071001xx s 000 0 eng 10.1007/s11139-007-9027-7 doi (NATIONALLICENCE)springer-10.1007/s11139-007-9027-7 Katsurada Masanori Department of Mathematics, Hiyoshi Campus, Keio University, 4-1-1 Hiyoshi, Kouhoku-ku, 223-8521, Yokohama, Japan aut Complete asymptotic expansions associated with Epstein zeta-functions [Elektronische Daten] [Masanori Katsurada] 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 $\zeta _{\mathbb {Z}^{2}}(s;z)=\sum_{m,n=-\infty}^{\infty}Q(m,n)^{-s}$ for Re s>1, where the term with m=n=0 is to be omitted. We deduce complete asymptotic expansions of $\zeta _{\mathbb {Z}^{2}}(s;x+iy)$ as y→+∞ (Theorem1 in Sect.2), and of its weighted mean value (with respect to y) in the form of a Laplace-Mellin transform of $\zeta _{\mathbb {Z}^{2}}(s;z)$ (Theorem2 in Sect.2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of $\zeta _{\mathbb {Z}^{2}}(s;z)$ over the whole s-plane is prepared by means of Mellin-Barnes integral transformations (Proposition1 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 $\zeta _{\mathbb {Z}^{2}}(s;z)$ (Proposition2 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. Springer Science+Business Media, LLC, 2007 Epstein zeta-function nationallicence Riemann zeta-function nationallicence Laplace-Mellin transform nationallicence Mellin-Barnes integral nationallicence Weighted mean value nationallicence Asymptotic expansion nationallicence The Ramanujan Journal Springer US; http://www.springer-ny.com 14/2(2007-10-01), 249-275 1382-4090 14:2<249 2007 14 11139 https://doi.org/10.1007/s11139-007-9027-7 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s11139-007-9027-7 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Katsurada Masanori Department of Mathematics, Hiyoshi Campus, Keio University, 4-1-1 Hiyoshi, Kouhoku-ku, 223-8521, Yokohama, Japan aut NATIONALLICENCE 773 0- The Ramanujan Journal Springer US; http://www.springer-ny.com 14/2(2007-10-01), 249-275 1382-4090 14:2<249 2007 14 11139 SWISSBIB 463178076