# An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-functions (II).

Collectanea Mathematica (2005)

- Volume: 56, Issue: 1, page 57-83
- ISSN: 0010-0757

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topKatsurada, Masanori. "An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-functions (II).." Collectanea Mathematica 56.1 (2005): 57-83. <http://eudml.org/doc/41821>.

@article{Katsurada2005,

abstract = {For the Lerch zeta-function Φ(s,x,λ) defined below, the multiple mean square of the form (1.1), together with its discrete and Irbid analogues, (1.2) and (1.3) are investigated by means of Atkinson's [2] dissection method applied to the product Φ(u,x,λ)Φ(υ,x,-λ), where u and υ are independent complex variables (see (4.2)). A complete asymptotic expansion of (1.1) as Im s → ±∞ is deduced from Theorem 1, while those of (1.2) and (1.3) as q → ∞ and (at the same time) as Im s → ±∞ are deduced from Theorems 2 and 3 respectively. In the proofs, Atkinson's method above is enhanced by Mellín-Barnes type of integral formulae (see (4.1)), which further enable us systematic use of various properties of hypergeometric functions (see Section 5); especially in the proof of Theorem 1 crucial roles are played by Lemmas 3 and 5. },

author = {Katsurada, Masanori},

journal = {Collectanea Mathematica},

keywords = {Teoría analítica de números; Función zeta; Media cuadrática; Desarrollo asintótico; Riemann zeta-function; Hurwitz zeta-function; Lerch zeta-function; Mellin-Barnes integral; mean square; asymptotic expansion},

language = {eng},

number = {1},

pages = {57-83},

title = {An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-functions (II).},

url = {http://eudml.org/doc/41821},

volume = {56},

year = {2005},

}

TY - JOUR

AU - Katsurada, Masanori

TI - An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-functions (II).

JO - Collectanea Mathematica

PY - 2005

VL - 56

IS - 1

SP - 57

EP - 83

AB - For the Lerch zeta-function Φ(s,x,λ) defined below, the multiple mean square of the form (1.1), together with its discrete and Irbid analogues, (1.2) and (1.3) are investigated by means of Atkinson's [2] dissection method applied to the product Φ(u,x,λ)Φ(υ,x,-λ), where u and υ are independent complex variables (see (4.2)). A complete asymptotic expansion of (1.1) as Im s → ±∞ is deduced from Theorem 1, while those of (1.2) and (1.3) as q → ∞ and (at the same time) as Im s → ±∞ are deduced from Theorems 2 and 3 respectively. In the proofs, Atkinson's method above is enhanced by Mellín-Barnes type of integral formulae (see (4.1)), which further enable us systematic use of various properties of hypergeometric functions (see Section 5); especially in the proof of Theorem 1 crucial roles are played by Lemmas 3 and 5.

LA - eng

KW - Teoría analítica de números; Función zeta; Media cuadrática; Desarrollo asintótico; Riemann zeta-function; Hurwitz zeta-function; Lerch zeta-function; Mellin-Barnes integral; mean square; asymptotic expansion

UR - http://eudml.org/doc/41821

ER -

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