Specific values of higher derivatives of zeta functions : zeta

1. Lerch ζ函数的高阶导数的具体值：通式中，s为复变量，a，λ为带>的实参数，φ(s,a,λ)为Dirichlet级数Σ^∞_<n=0>e^<2πin>(n+a)^<-s>所定义的Lerch ζ函数及其在整个s平面上的亚纯延展。在本研究的前期，主要研究者通过扇形|arg z| < π建立了φ(s,a+z,λ)为z→∞的完全渐近展开式，并将其进一步应用于研究高阶导数R_<k,m>(z,λ) = (-1)^<k+1>(δ/δs)^kφ(s,z,λ)|_<s=-m> （k,m = 0,1，…）的特殊值。得到R_<k,m>(z,λ)的Taylor展开式，即Gauβ、weerstraβ和Plana型的公式；这些与证明一起组织在论文“与lach ζ函数相关的幂级数和渐近级数：在高阶导数中的应用”中（提交前准备的预印本）。更多的函数：Hurwitz ζ函数ζ(s,1 + x)，一个特殊的例子，是通过在黎曼ζ函数ζ(s) = ζ(s,1)的每个求和中移动n为n + x （n = 1,2，…）得到的。首席调查员最近在《收集》中概括了他以前的研究结果。数学。48(1997)]，证明了均方∫^1_0|φ(s,1+x,λ)|^2dx在t = ims→±∞时的完全渐近展开式，证明了多个均方∫^1_0【三键】∫^1_0|φ(s，a+x_1+【三键】+x_m，λ)|^2dx_1【三键】dx_m （a >：一个常数；m = 1,2，…）仍然存在一个类似的渐近级数。这些结果及其证明被组织在论文“Mellin-Barnes型积分在Lerch ζ函数均方中的应用II”中（提交出版）。Epstein ζ函数及其积分变换：设z = x + y为复上半平面上的参数。然后，以二次形式Q(u,v) = |u + vz|^2的Epstein ζ函数ζ_<Z^2>（s； Z）定义为ζ_<Z^2>(s； Z) = Σ ^</∞>_<m,n=-∞>Q(m,n)^<-s>（省略m = n= 0的项）及其在整个s平面上的亚纯延拓；这在（算术）二次型的研究中起着重要作用。最近，首席研究员建立了ζ_< z ^2>(s;z)的完全渐近展开式y = imz→+∞及其Laplace-Mellin变换（可视为泊松分布权值的均值）。得到的结果与他们的证明被组织在论文“与Epstein - ζ函数相关的完全渐近展开式”中（提交发表）。少

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