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

• Autores: Masanori Katsurada
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 56, Fasc. 1, 2005, págs. 57-83
• Idioma: inglés
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• Resumen

  • For the Lerch zetafunction _(s, x, _) 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 _(u, x, _)_(v, x,-_), where u and v are independent complex variables (see (4.2)). A complete asymptotic expansion of (1.1) as Im s ! ±1 is deduced from Theorem 1, while those of (1.2) and (1.3) as q ! 1and (at the same time) as Im s ! ±1 are deduced from Theorems 2 and 3 respectively. In the proofs, Atkinson's method above is enhanced by MellinBarnes 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 r¿oles are played by Lemmas 3 and 5.