Behaviour of zeta and theta functions : their intrinsic linkage

1 . Lerch ζ函数的多重均方：设s为复变量，x和a为实参数，x > 0，记为e(λ)=e^<2π λ>。Lerch ζ函数φ(s, x, λ)定义为级数Σ^∞_<l=0> e(λl)(l+x)^<-5>，及其在整个s平面上的亚纯延拓；当λ∈Σ时，这简化为Hurwitz ζ函数ζ(s, x)，并进一步简化为黎曼ζ函数ζ(s)=ζ(s, 1)。注意，定义级数ζ(s, 1+x)是通过将ζ(s)的每一项从l移到1+x （l= 1,2，…）得到的。设m >为任意整数，a >为固定实数。在此背景下，首席研究员引入并研究了形式为∫^1_0【三键】∫^1_0|φ(s， a + x_1 +【三键】+ x_m，λ)|^2dx_1【三键】dx_m的多重均方，并通过改进其先前研究[Collect]中开发的方法，建立了其ims→±∞的完全渐近展开式。数学。（1997）)出现在[文集]中。数学。(2005)]) .II。非纯纯爱森斯坦级数的完全渐近展开式：设z = x + y在复上半平面上。具有二次型Q(u, v)=|u+vz|^2的Epstein ζ函数ζ_<z^2> (s; z)被定义为级数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；当y = Im z→+∞时，ζz^2 （s; z）的渐近方面起着至关重要的作用，例如在二次型的算术研究中。最近，首席研究者建立了ζz^2 （s; z）为→+∞的完全渐近展开，并进一步证明了ζz^2 （s; z）的Laplace-Mellin变换沿z的虚方向为→+∞仍然存在一个类似的渐近级数（出现在[Ramanujan J.]中）。接下来设κ为任意偶数。然后用级数（y^s/2） Σ_<c, d=1>(cz + d)^<-κ> |cz + d|^<-2_s>及其在整个s平面上的亚纯延展来定义SL_2(Z)上的非全纯Eisenstein级数（权值为κ）；这表明当κ = 0时，关系E_0(s; z) = y^s ζz^2 (s; z)/2ζ(2s)，这很容易得到E_0（s; z）作为y→+∞的完全渐近展开式。最近，首席研究员（与日本大学的T. Noda教授合作）在将上面的E_0（s; z）渐近展开式转化为E_κ(s; z)渐近展开式之后，通过连续使用Maass的权变算子，建立了E_κ(s; z)为y→+∞的任意偶数k的完全渐近展开式（已提交发表）。少

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