论文标题
限制随机dirichlet系列的定理
Limit theorems for random Dirichlet series
论文作者
论文摘要
我们证明了在随机dirichlet系列$ d（α; z）= \ sum_ {n \ geq 2}（\ log n）^α（η_n+{\ rm i}θ_n）/n^z $中，适当缩放并正常化的随机dirichlet系列$ d（α; z）= \ sum_ {n \ geq 2} = \ sum_ {n \ geq 2} = \ sum_ {n \ geq 2}（正确） $（η_n，θ_n）_ {n \ in \ mathbb {n}} $是一个中心$ \ Mathbb {r}^2 $ valued andural vector $（η，θ）$的独立副本的序列，具有有限的第二瞬间和$α> -1/2 $是一个固定的实际参数。结果，我们表明，复杂和真实零的点过程$ d（α; z）$含糊地收敛，从而获得了普遍性结果。在实际情况下，也就是说，当$ \ mathbb {p} \ {θ= 0 \} = 1 $时，我们还证明了对$ d（α; z）$的迭代对数定律，已正确归一化，为$ z \ to（1/2）+$。
We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series $D(α;z)=\sum_{n\geq 2}(\log n)^α(η_n+{\rm i} θ_n)/n^z$, properly scaled and normalized, where $(η_n,θ_n)_{n\in\mathbb{N}}$ is a sequence of independent copies of a centered $\mathbb{R}^2$-valued random vector $(η,θ)$ with a finite second moment and $α>-1/2$ is a fixed real parameter. As a consequence, we show that the point processes of complex and real zeros of $D(α;z)$ converge vaguely, thereby obtaining a universality result. In the real case, that is, when $\mathbb{P}\{θ=0\}=1$, we also prove a law of the iterated logarithm for $D(α;z)$, properly normalized, as $z\to (1/2)+$.
