论文标题
涉及一些含义$ξ（t）$的某些积分的广义digamma函数和渐近学的模块化关系
A modular relation involving a generalized digamma function and asymptotics of some integrals containing $Ξ(t)$
论文作者
论文摘要
$ f（α，w）= f（β，iw）$的模块化关系，其中$ i = \ sqrt {-1} $和$αβ= 1 $。它涉及广义digamma函数$ψ_W（a）$，该$最近是由作者在开发广义Hurwitz Zeta函数$ζ_W（s，a）$的工作中研究的。该模块化关系的限制案例$ w \ to0 $是Ramanujan在Page $ 220 $ 220 $丢失笔记本的著名结果。我们还获得了涉及Riemann函数$ξ（t）$的一般积分的渐近估计，为$α\ to \ infty $。它不仅给出了我们的模块化关系中积分的渐近估计值，作为推论，而且还有一些已知的结果。
A modular relation of the form $F(α, w)=F(β, iw)$, where $i=\sqrt{-1}$ and $αβ=1$, is obtained. It involves the generalized digamma function $ψ_w(a)$ which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function $ζ_w(s, a)$. The limiting case $w\to0$ of this modular relation is a famous result of Ramanujan on page $220$ of the Lost Notebook. We also obtain asymptotic estimate of a general integral involving the Riemann function $Ξ(t)$ as $α\to\infty$. Not only does it give the asymptotic estimate of the integral occurring in our modular relation as a corollary but also some known results.
