论文标题
Dirichlet $ L $ functions的显式子概念估算
Explicit Subconvexity Estimates for Dirichlet $L$-functions
论文作者
论文摘要
给定一个dirichlet字符$χ$ modulo $ q $及其关联的$ l $ function，$ l（s，χ）$，我们提供了$ | l（s，χ）| $的burgess'估计的明确版本。我们使用部分求和来沿垂直线$ \ re {s} = 1 - {r}^{ - 1} $提供界限，其中$ r $是与Burgess的字符总和估算相关的参数。然后，使用Phragmén--lindelöf原理将这些边界连接到临界条上。特别是，对于$σ\在[\ frac {1} {2}，\ frac {9} {10}] $中，我们建立 $$ | l（σ+ it，χ）| \ leq（1.105）（0.692）^σQ^{\ frac {31} {80} {80} - \ frac {2} {5} {5}σ}（\ log {q}）^{\ frac {\ frac {33}
Given a Dirichlet character $χ$ modulo $q$ and its associated $L$-function, $L(s,χ)$, we provide an explicit version of Burgess' estimate for $|L(s, χ)|$. We use partial summation to provide bounds along the vertical lines $\Re{s} = 1 - {r}^{-1}$, where $r$ is a parameter associated with Burgess' character sum estimate. These bounds are then connected across the critical strip using the Phragmén--Lindelöf principle. In particular, for $σ\in [\frac{1}{2}, \frac{9}{10}]$, we establish $$|L(σ+ it, χ)| \leq (1.105) (0.692)^σq^{\frac{31}{80}-\frac{2}{5}σ}(\log{q})^{\frac{33}{16}-\frac{9}{8}σ} |σ+ it|.$$
