本研究代表者は，i) Lerch ゼータ関数の主変数 $s$ に関する Laplace-Mellin 型，Riemann-Liouville 型積分変換，及び，それらの適切な iteration(s) に対して，$s$ が扇状領域 $|arg s|<pi$ 内を其々 $sto0$ 及び $stoinfty$ となるときの完全漸近展開を確立した；ii)一般化正則 Eisenstein 級数に対して，付随するパラメタ $z$ が複素上半平面 $0<arg z<pi$ 内を $zto0$ 及び $zto iinfty$ となる其々の場合について，完全漸近展開を確立した．
i) Averaging of the values of Lerch zeta-functions: The head investigator has shown that complete asymptotic expansions exist for the Laplace-Mellin and Riemann-Liouville transforms, together with their appropriate iterations, of Lerch zeta-functions in terms of their pivotal variable $s$ of the transforms, when $sto0$ and $stoinfty$ both through the sector $|arg s|<pi$. The region of validity of these asymptotic expansions contain any vertical ray through imaginary directions; this allows us in general fairly nice applicability to the problems of analytic number theory;
ii) Asymptotic expansions associated with Dirichlet-Hurwitz-Lerch holomorphic Eisenstein series: The head investigator, jointed with (his collaborator) Professor Takumi Noda, have established complete asymptotic expansions exist for Dirichlet-Hurwitz-Lerch holomorphic Eisenstein series when the associated parameter $z$ of the series tends to $0$ and $infty$ both through the complex upper half-plane $0<arg z<pi$.

研究種目 : 基盤研究 (C) (一般)
研究期間 : 2017～2021
課題番号 : 17K05182
研究分野 : 解析的整数論