Behaviour of zeta and theta functions : their intrinsic linkage

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I. Multiple mean square of Lerch zeta-functions : Let s be a complex variable, x and A be real parameter with x > 0, and write e(λ)=e^<2πiλ>. The Lerch zeta-function φ(s, x, λ) is defined by the series Σ^∞_<l=0> e(λl)(l+x)^<-5>, and its meromorphic continuation over the whole s-plane; this reduces to the Hurwitz zeta-function ζ(s, x) when λ ∈ Σ, and further to the Riemann zeta-function ζ(s)=ζ(s, 1). Note that the defining series ζ(s, 1+x) is obtained by shifting each term of ζ(s) as l to 1 + x(l=1, 2,...). Let m > 1 be any integer, and a > 0 a fixed real number. In this context the head investigator introduced and studied a multiple mean square of the form ∫^1_0【triple bond】∫^1_0|φ(s, a + x_1 +【triple bond】 + x_m,λ)|^2dx_1【triple bond】dx_m, for which a complete asymptotic expansion as Im s →±∞ has been established by refining the method developed in his previous study [Collect. Math. (1997)] (appeared in [Collect. Math. (2005)]).II. Complete asymptotic expansions associated with non-ho … More lomorphic Eisenstein series : Let z = x + iy be in the complex upper-half plane. The Epstein zeta-function ζ_<z^2> (s; z), attached to the quadratic form Q(u, v)=|u+vz|^2, is defined by the series Σ^∞_<m, n=-∞>Q(m, n)^<-s> (upon omitting the term with m=n=0), and its meromorphic continuation over the whole s-plane; the asymptotic aspects of ζz^2 (s; z) as y = Im z→ +∞ play crucial roles, for e.g., in arithmetical study of quadratic forms. The head investigator recently established a complete asymptotic expansion of ζz^2 (s; z) as → +∞, the proof of which was further elaborated to show that a similar asymptotic series still exists for the Laplace-Mellin transform of ζz^2 (s; z) along the imaginary direction of z as →+∞ (to appear in [Ramanujan J.]). Next let κ be any even integer. Then the non-holomorphic Eisenstein series (of weight κ) attached to SL_2(Z) is defiend by the series(y^s/2) Σ_<c, d=1>(cz + d)^<-κ> |cz + d|^<-2_s>, and its meromorphic continuation over the whole s-plane; this shows when κ = 0 the relation E_0(s; z) = y^s ζz^2 (s; z)/2ζ(2s), which readily yields a complete asymptotic expansion of E_0(s; z) as y →+∞. The head investigator recently established (jointly with Prof. T. Noda at Nihon Univ.) a complete asymptotic expansion of E_κ(s; z) as y →+∞ for any even integer k through the successive use of Maass' weight change operators, upon transferring from the asymptotic expansion of E_0(s; z) above to that of E_κ(s; z) (submitted for publication). Less