### Abstract

Let Q(u, v) = |u + vz|<sup>2</sup> be a positive-definite quadratic form with a complex parameter z = x +iy in the upper half-plane. The Epstein zeta-function ζZ<sup>2</sup> (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 ζZ<sup>2</sup> (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 ψZ<sup>2</sup> (s; a, b; μ, ν; z) defined by (1.2) below (Theorem 1), and also for the Riemann–Liouville transform (1.5) of ζZ<sup>2</sup> (s; z) (Theorem 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of ψZ<sup>2</sup> (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 ψZ<sup>2</sup> (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.

Original language | English |
---|---|

Pages (from-to) | 403-437 |

Number of pages | 35 |

Journal | Ramanujan Journal |

Volume | 36 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2015 |

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### Keywords

- Asymptotic expansion
- Epstein zeta-function
- Mellin–Barnes integral
- Riemann–Liouville transform
- Weighted mean value

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Complete asymptotic expansions associated with Epstein zeta-functions II.** / Katsurada, Masanori.

Research output: Contribution to journal › Article

*Ramanujan Journal*, vol. 36, no. 3, pp. 403-437. https://doi.org/10.1007/s11139-014-9583-6

}

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 -