TY - JOUR

T1 - Complete asymptotic expansions associated with Epstein zeta-functions II

AU - Katsurada, Masanori

PY - 2015

Y1 - 2015

N2 - 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 ζZ2 (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 ζZ2 (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 ψZ2 (s; a, b; μ, ν; z) defined by (1.2) below (Theorem 1), and also for the Riemann–Liouville transform (1.5) of ζZ2 (s; z) (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 (Proposition 1 in Sect. 3). This procedure differs slightly from other previously known methods of analytic continuation, and provides the meromorphic continuation of ψZ2 (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.

AB - 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 ζZ2 (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 ζZ2 (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 ψZ2 (s; a, b; μ, ν; z) defined by (1.2) below (Theorem 1), and also for the Riemann–Liouville transform (1.5) of ζZ2 (s; z) (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 (Proposition 1 in Sect. 3). This procedure differs slightly from other previously known methods of analytic continuation, and provides the meromorphic continuation of ψZ2 (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.

KW - Asymptotic expansion

KW - Epstein zeta-function

KW - Mellin–Barnes integral

KW - Riemann–Liouville transform

KW - Weighted mean value

UR - http://www.scopus.com/inward/record.url?scp=84939894285&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939894285&partnerID=8YFLogxK

U2 - 10.1007/s11139-014-9583-6

DO - 10.1007/s11139-014-9583-6

M3 - Article

AN - SCOPUS:84939894285

VL - 36

SP - 403

EP - 437

JO - The Ramanujan Journal

JF - The Ramanujan Journal

SN - 1382-4090

IS - 3

ER -