TY - JOUR

T1 - Complete asymptotic expansions associated with Epstein zeta-functions

AU - Katsurada, Masanori

N1 - Funding Information:
Research supported in part by Grant-in-Aid for Scientific Research (No. 13640041), the Ministry of Education, Culture, Sports, Science and Technology of Japan.

PY - 2007/10

Y1 - 2007/10

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 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.

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 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.

KW - Asymptotic expansion

KW - Epstein zeta-function

KW - Laplace-Mellin transform

KW - Mellin-Barnes integral

KW - Riemann zeta-function

KW - Weighted mean value

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U2 - 10.1007/s11139-007-9027-7

DO - 10.1007/s11139-007-9027-7

M3 - Article

AN - SCOPUS:34648816153

VL - 14

SP - 249

EP - 275

JO - The Ramanujan Journal

JF - The Ramanujan Journal

SN - 1382-4090

IS - 2

ER -