The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form \((\mu \star \Lambda _1^{\star k_1} \star \Lambda _2^{\star k_2} \star \cdots \star \Lambda _d^{\star k_d})\) is computed unconditionally by means of the autocorrelation of ratios of \(\zeta \) techniques from Conrey et al. (Proc Lond Math Soc (3) 91:33–104, 2005), Conrey et al. (Commun Number Theory Phys 2:593–636, 2008) as well as Conrey and Snaith (Proc Lond Math Soc 3(94):594–646, 2007). This in turn allows us to describe the combinatorial process behind the mollification of$$\begin{aligned} \zeta (s) + \lambda _1 \frac{\zeta '(s)}{\log T} + \lambda _2 \frac{\zeta ''(s)}{\log ^2 T} + \cdots + \lambda _d \frac{\zeta ^{(d)}(s)}{\log ^d T}, \end{aligned}$$where \(\zeta ^{(k)}\) stands for the kth derivative of the Riemann zeta-function and \(\{\lambda _k\}_{k=1}^d\) are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (Res Number Theory 4:9, 2018), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

中文翻译：

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$$ \ zeta $$ζ的零点中超过五分之二的位置在临界线上

Riemann zeta函数的第二阶被规范化Dirichlet多项式扭曲，其系数形式为\（（\ mu \ star \ Lambda _1 ^ {\ star k_1} \ star \ Lambda _2 ^ {\ star k_2} \ star \通过Conrey等人的\（\ zeta \）技术比率的自相关，无条件计算cdots \ star \ Lambda _d ^ {{star k_d}）\）。（Proc Lond Math Soc（3）91：33–104，2005），Conrey等。（Commun Number Theory Phys 2：593–636，2008）以及Conrey和Snaith（Proc Lond Math Soc 3（94）：594–646，2007）。反过来，这又使我们能够描述放松情绪背后的组合过程。$$ \ begin {aligned} \ zeta（s）+ \ lambda _1 \ frac {\ zeta'（s）} {\ log T} + \ lambda _2 \ frac {\ zeta''（s）} {\ log ^ 2 T} + \ cdots + \ lambda _d \ frac {\ zeta ^ {{d}} {s}} {\ log ^ d T}，\ end {aligned} $$其中\（\ zeta ^ {（k） } \表示黎曼zeta函数的第k个导数，并且\（\ {\ lambda _k \} _ {k = 1} ^ d \）是实数。作为对应用的改进，我们改进了最近的长动词和Pratt和Robles导致的Kloosterman总和的结果（Res Number Theory 4：9，2018），将当前的黎曼zeta函数临界零的下限提高到略大于五分之二。