Specific values of higher derivatives of zeta functions : zeta

zeta 函数高阶导数的具体值：zeta

• 批准号: 13640041
• 负责人: KATSURADA Masanori
• 金额: $ 2.11万
• 依托单位: Keio University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640041/
• 关键词: zeta function; higher derivative; specific value; asymptotic expansion; メリン変換

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

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

项目成果

M.Amou: "On the values of certain q-hypergeometric series II"in Analytic Number Theory, Kluver, Dev. Math.. 6. 17-25 (2002)

M.Amou：《解析数论中某些 q 超几何级数 II 的值》，Kluver，Dev。

M.Amou, M.Katsurada, K.Vaananen: "On the values of certain q-hypergeometric series"in "Number Theory : Proc.Turku Symp.Number Theory, "M.Jutila and T.Matsankyla (Eds.) de Gruyter. 5-17 (2001)

M.Amou、M.Katsurada、K.Vaananen：《数论：Proc.Turku Symp.Number Theory》中的“论某些 q 超几何级数的值”，M.Jutila 和 T.Matsankyla (Eds.) de Gruyter

Katsurada, M.: "Asymptotic expansions of certain q-series and a formula for specific values of the Riemann zeta-function"Acta Arithmetica. 107・3. 269-298 (2003)

Katsurada, M.：“某些 q 级数的渐近展开式和黎曼 zeta 函数的特定值的公式”Acta Arithmetica 107・3（2003）。

S.Tsuboi: "The Euler number of the normalization of an algebraic threefold with ordinary singularities""Geometric Singularity Theory", Banach Center Publications, Polish Academy of Sciences, Warzawa. (To appear). (2004)

S.Tsuboi：“具有普通奇点的代数三重标准化的欧拉数”“几何奇点理论”，巴纳赫中心出版物，波兰科学院，瓦尔扎瓦。

M.Katsurada, K.Matsumoto: "Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions III"Compositio Math.. 131. 239-266 (2002)

M.Katsurada、K.Matsumoto：“Hurwitz zeta 函数 III 的某些均方的显式公式和渐近展开”Compositio Math.. 131. 239-266 (2002)