论文标题
$ζ$的100％的零位于关键线上
100% of the zeros of $ζ(s)$ are on the critical line
论文作者
论文摘要
在整个手稿中，将零以多重性计数。我们用$ n（t）$表示关键条件中的$ρ$ $ζ$的数量，直至高度$ t $，其中$ t> 3 $不是$ζ（s）$的零坐标。用$ n_0（t）$表示$ρ$ $ζ（s）$的零$ρ$的数量，直至高度$ t $。我们首先表明存在$ε_0> 0 $，因此$ξ（s）$在一个小矩形$r_ε$的边界上没有零，定义为$r_ε= \ {σ+it \ in \ mathbb {c} c} \ frac {1} {2}+ε，\ 0 \ leq t \ leq t \} $每当$ 0 <ε<ε_0$时。其次，如果$n_ε（t）$是矩形$r_ε$内部的$ρ$ $ρ$的数量，那么根据高度$ t $，我们证明$n_ε（t）= n_0（t）$ for $ε$足够小。我们在矩形$r_ε$上使用Littlewood的引理以及$ξ（s）$的Hadamard产品以及$ζ（S）$的对数衍生产品的渐近线$ n_0（t）= \ frac {t} {2π} \ log \ left（\ frac {t} {t} {2π} \ right） - \ frac {t} {2π}+\ nathcal {othcalcal {o}（\ log t）$ $κ：= \ liminf_ {t \ to \ infty} \ frac {n_0（t）} {n（t）} $$，因此我们证明$κ= 1 $。
Throughout this manuscript the zeros are counted with multiplicity. We denote by $N(T)$ the number of zeros $ρ$ of $ζ(s)$ in the critical strip upto height $T$ where $T>3$ is not an ordinate of zero of $ζ(s)$. Denote by $N_0(T)$ the number of zeros $ρ$ of $ζ(s)$ on the critical line upto height $T$. We first show that there exists $ε_0>0$ such that $ξ(s)$ has no zeros on the boundary of a small rectangle $R_ε$ defined as $R_ε=\{σ+it\in\mathbb{C}\mid \frac{1}{2}-ε\leq σ\leq \frac{1}{2}+ε,\ 0\leq t\leq T\}$ whenever $0<ε<ε_0$. Secondly if $N_ε(T)$ is the number of zeros $ρ$ of $ζ(s)$ inside the rectangle $R_ε$ then we prove that $N_ε(T)=N_0(T)$ for $ε$ sufficiently small depending on the height $T$. We use the Littlewood's lemma on the rectangle $R_ε$ along with the Hadamard product of $ξ(s)$ and the asymptotic for the logarithmic derivative of $ζ(s)$ to prove that as $T\to \infty$, $$N_0(T)=\frac{T}{2π}\log\left(\frac{T}{2π}\right)-\frac{T}{2π}+\mathcal{O}(\log T)$$ Also if $κ$ is the proportion of zeros of $ζ(s)$ on the critical line $$κ:=\liminf_{T\to \infty} \frac{N_0(T)}{N(T)}$$ then we prove as a consequence that $κ=1$.
