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

For the Lerch zeta-function $\phi(s, x, \lambda)$ defined below, the multiple mean square of the form (1.1), together with its discrete and hybrid analogues, (1.2) and (1.3), are investigated by means of Atkinson's [2] dissection method applied to the product $\phi(u, x, \lambda) \phi (v, x, -\lambda)$, where $u$ and $v$ are independent complex variables (see (4.2)). A complete asymptotic expansion of (1.1) as Im $s\rightarrow\pm\infty$ is deduced from Theorem 1, while those of (1.2) and (1.3) as $q\rightarrow\infty$ and (at the same time) as Im $s\rightarrow\pm\infty$ are deduced from Theorems 2 and 3 respectively. In the proofs, Atkinson's method above is enhanced by Mellin-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 rôles are played by Lemmas 3 and 5.