このアイテムのアクセス数: 97
このアイテムのファイル:
ファイル | 記述 | サイズ | フォーマット | |
---|---|---|---|---|
2203-06.pdf | 1.75 MB | Adobe PDF | 見る/開く |
タイトル: | COMPLETE ASYMPTOTIC EXPANSIONS FOR THE TRANSFORMED LERCH ZETA-FUNCTIONS VIA THE LAPLACE-MELLIN AND RIEMANN-LIOUVILLE OPERATORS (PRE-ANNOUNCEMENT) (Problems and prospects in Analytic Number Theory) |
著者: | KATSURADA, MASANORI |
著者名の別形: | 桂田, 昌紀 |
キーワード: | 11M35 11M06 Lerch zeta-function Laplace-Mellin Transform Riemann-Liouville transform Mellin-Barnes integral asymptotic expansion power series expansion weighted mean value |
発行日: | Nov-2021 |
出版者: | 京都大学数理解析研究所 |
誌名: | 数理解析研究所講究録 |
巻: | 2203 |
開始ページ: | 56 |
終了ページ: | 67 |
抄録: | This is a pre-announcement version of the forthcoming paper [Complete asymptotic expansions for the transformed Lerch zeta-functions via the LaplaceMellin and Riemann-Liouville operators, preprint.]. For a complex variable s, and any real parameters a and λ with a > 0, the Lerch zeta-function φ(s, a, λ) is defined by the Dirichlet series Σ[∞][l=0]e(λl)(a + l)⁻[s] (Res> 1), and its meromorphic continuation over the whole s-plane, where e(λ) = e[2πiλ], and the domain of the parameter a can be extended to the whole sector |arg z| < π through the procedure in [M. Katsurada, Power series and asymptotic series associated with the Lerch zeta-function, Proc. Japan Acad. Ser. A Math. Sci. 74 (1998), 167-170.]. It is the principal aim of the present article to treat asymptotic aspects of the transformed functions obtained by applying the Laplace-Mellin and Riemann-Liouville operators (in terms of the variables), which are denoted by LM[α][z;T] and RL[α, β][z;T] respectively, to a slight modification, φ*(s, a, λ), of φ(s, a, λ). For any m ∈ ℤ, let (φ*)[(m)](s, a, λ) denote the mth derivative with respect to s if m ≥ 0, and the |m|th primitive defined with its initial point at s + ∞ if m < 0. We shall then show that complete asymptotic expansions exist, if a > 1, for .LM[α][z;T](φ*)[(m)](s+τ, a, λ) and for RL[α, β][z;T](φ*)[(m)](s+τ, a, λ) (Theorems 1-4), as well as for their severa.l iterated variants (Theorems 5-10), when the pivotal parameter z of the transforms tends to both O and oo through appropriate sectors. Most of our results include any vertical ha.If-lines in their respective regions of validity; this allows us to deduce complete asymptotic expansions for the relevant transforms through arbitrary vertical half-lines, upon taking (s, z) = (a, it) with any σ ∈ ℝ, when t → ±∞ (Corollaries 2.1, 4.1, 6.1 and 8.1). |
URI: | http://hdl.handle.net/2433/267788 |
出現コレクション: | 2203 解析的整数論の諸問題と展望 |

このリポジトリに保管されているアイテムはすべて著作権により保護されています。