扰动时刻和$ζ$临界零点的较长缓和剂
Perturbed moments and a longer mollifier for critical zeros of $ζ$
摘要：
设$A(s)$是一般的Dirichlet多项式，$Φ$是支持在$[1,2]$中的光滑函数，其导数有弱界.给出了积分$I(α,β)=\int_{\mathbb{R}} ζ(\frac{1}{2}+α+it)ζ(\frac{1}{2}+β+it)|A(\frac{1}{2}+it)|^2 Φ(\frac{t}{T})dt$的新的主要项。对于误差项，通过将误差项分解为I型和II型和，然后研究Kloosterman和的结果，证明了Feng软化子的长度可以从$θ< \frac{17}{33}$增加到$θ< \frac{6}{11}$.作为应用，我们略微增加了$ζ(s)$在临界线上的零的比例。
英文摘要：
Let $A(s)$ be a general Dirichlet polynomial and $Φ$ be a smooth function supported in $[1,2]$ with mild bounds on its derivatives. New main terms for the integral $I(α,β)=\int_{\mathbb{R}} ζ(\frac{1}{2}+α+it)ζ(\frac{1}{2}+β+it)|A(\frac{1}{2}+it)|^2 Φ(\frac{t}{T})dt$ are given. For the error term, we show that the length of the Feng mollifier can be increased from $θ< \frac{17}{33}$ to $θ< \frac{6}{11}$ by decomposing the error into Type I and Type II sums and then studying the resulting sums of Kloosterman sums. As an application, we slightly increase the proportion of zeros of $ζ(s)$ on the critical line.
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