论文标题
计数dirichlet $ l $ functions的零
Counting Zeros of Dirichlet $L$-Functions
论文作者
论文摘要
我们给出了$ n（t，χ）$的明确上限和下限，这是dirichlet $ l $ function的零零的数量，最多$χ$和高度$ t $。假设$χ$具有导体$ q> 1 $，而$ t \ geq 5/7 $。如果$ \ ell = \ log \ frac {q（t+2）} {2π}> 1.567 $，则\ begin {equation*} \ left | n（t，χ） - \ left（\ frac {t}π\ log \ frac {qt} {2πe} - \ frac {χ（-1）} {4} {4} \ right）\ right | \ le 0.22737 \ ell + 2 \ log（1+ \ ell）-0.5。 \ end {等式*}我们给出小$ q $和$ t $的结果稍强。一路上，我们证明了$ | l（s，χ）| $ $σ<-1/2 $的新结合。
We give explicit upper and lower bounds for $N(T,χ)$, the number of zeros of a Dirichlet $L$-function with character $χ$ and height at most $T$. Suppose that $χ$ has conductor $q>1$, and that $T\geq 5/7$. If $\ell=\log\frac{q(T+2)}{2π}> 1.567$, then \begin{equation*} \left| N(T,χ) - \left( \frac{T}π \log\frac{qT}{2πe} -\frac{χ(-1)}{4}\right) \right| \le 0.22737 \ell + 2 \log(1+\ell) - 0.5. \end{equation*} We give slightly stronger results for small $q$ and $T$. Along the way, we prove a new bound on $|L(s,χ)|$ for $σ<-1/2$.
