# 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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## Abstract

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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.

## How to cite

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Katsurada, 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>.

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. },
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
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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