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.
@article{urn:eudml:doc:41821,
title = {An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-functions (II).},
journal = {Collectanea Mathematica},
volume = {56},
year = {2005},
pages = {57-83},
zbl = {1059.11051},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41821}
}
Katsurada, Masanori. An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-functions (II).. Collectanea Mathematica, Tome 56 (2005) pp. 57-83. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41821/