Specific values of higher derivatives of zeta functions : zeta

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1. Specific values of higher derivatives of the Lerch zeta-function : Throughout the following, s is a complex variable, a, λ are real parameters with a > 0, and φ(s,a,λ) denotes the Lerch zeta-function defined by the Dirichlet series Σ^∞_<n=0>e^<2πin>(n+a)^<-s>, and its meromorphic continuation over the whole s-plane. At an earlier stage of the present research, the head investigator established a complete asymptotic expansion of φ(s,a+z,λ) as z → ∞ through the sector |arg z| < π, which was further applied to study the particular values of higher derivatives R_<k,m>(z,λ) = (-1)^<k+1>(δ/δs)^kφ(s,z,λ)|_<s=-m> (k,m = 0,1,...). The results obtained for R_<k,m>(z,λ) are its Taylor expansion, the formulae of the types of Gauβ, Weierstraβ and Plana ; those together with the proofs are organized in the paper "Power series and asymptotic series associated with the Lerch zeta-function : applications to higher derivatives," (preprint prepared for submission).2. A multiple mean square of Lerch ze … More ta-functions : The Hurwitz zeta-function ζ(s,1 + x), a particular case of the Lerch zeta-function, is obtained by shifting n for n + x (n = 1,2,...) in each summand of the Riemann zeta-function ζ(s) = ζ(s,1). The head investigator recently generalized his previous result in [Collect. Math. 48 (1997)], which asserts a complete asymptotic expansion of the mean square ∫^1_0|φ(s,1+x,λ)|^2dx as t = Im s → ±∞, to show that a similar asymptotic series still exists for the multiple mean square ∫^1_0【triple bond】∫^1_0|φ(s,a+x_1+【triple bond】+x_m,λ)|^2dx_1【triple bond】dx_m (a > 0: a constant; m = 1,2,...). The results and their proofs are organized in the paper "An application of Mellin-Barnes type of integrals to the mean square of Lerch zeta-functions II," (submitted for publication).3. Epstein zeta-functions and their integral transforms : Let z = x + iy be a parameter in the complex upper half-plane. Then the Epstein zeta-function ζ_<Z^2>(s;z), attached to the quadratic form Q(u,v) = |u + vz|^2, is defined by ζ_<Z^2>(s;z) = Σ ^</∞>_<m,n=-∞>Q(m,n)^<-s> (upon omitting the term with m = n = 0), and its meromorphic continuation over the whole s-plane ; this plays an important role in the study of (arithmetical) quadratic forms. The head investigator recently established complete asymptotic expansions, as y = Im z → +∞, of ζ_<Z^2>(s;z) and its Laplace-Mellin transform (which can be regarded as a mean value with the weight of Poisson distribution). The results obtained are organized with their proofs in the paper "Complete asymptotic expansions associated with the Epstein zeta-function," (submitted for publication). Less