带一般权重的随机狄利克雷级数的重对数律
Laws of the iterated logarithm for random Dirichlet series with general weights
AI总结：
本文研究带一般权重的随机狄利克雷级数的重对数律，在特定假设下构造非标准LIL例子，证明正则增长条件不足，附加计数条件下得到几乎必然簇集[-1,1]并应用于数论系数序列。
AI中文摘要：
对于每个s>0，我们考虑随机狄利克雷级数X(s)=∑_{k≥1}k^{-1/2-s}a_kη_k，其中η₁,η₂,…是均值为零、有限正方差的独立同分布随机变量，(a_k)_{k≥1}是满足对每个s>0有∑_{k≥1}k^{-1-2s}a_k²<∞且∑_{k≥1}k^{-1}a_k²=∞的确定实数列。我们研究当s→0+时X(s)的几乎必然波动。在这些最小假设下，我们构造了沿合适序列呈现多种非标准重对数律（LIL）形式的例子：归一化因子及上下极限常数可能与经典对应项不同。我们还证明，形如∑_{k≤n}k^{-1}a_k²~c(log n)^β（其中c,β>0）的正则增长条件本身不足以确保标准LIL。最后，在控制权重a_k相对较大的指标出现频率的附加计数条件下，我们证明当s→0+时，(2Var[X(s)]loglog(Var[X(s)]))^{-1/2}X(s)的几乎必然簇集为[-1,1]。该结果被应用于几个数论来源的系数序列。
英文摘要：
For each $s>0$, we consider a random Dirichlet series $X(s)=\sum_{k\geq 1}k^{-1/2-s}a_kη_k$, where $η_1$, $η_2,\ldots$ are independent and identically distributed random variables with mean zero and finite positive variance, and $(a_k)_{k\geq 1}$ is a deterministic sequence of real numbers satisfying $\sum_{k\geq 1}k^{-1-2s}a_k^2<\infty$ for each $s>0$ and $\sum_{k\geq 1}k^{-1}a_k^2=\infty$. We investigate the almost-sure fluctuations of $X(s)$ as $s\to0+$. Under these minimal assumptions, we construct examples exhibiting several non-standard forms of the law of the iterated logarithm (LIL) along suitable sequences: the normalization and the upper and lower limit constants may differ from their classical counterparts. We also show that a regular growth condition of the form $\sum_{k\leq n}k^{-1}a_k^2\sim c(\log n)^β$, where $c,β>0$, is not by itself sufficient to ensure a standard LIL. Finally, under an additional counting condition controlling the frequency of indices at which the weights $a_k$ are comparatively large, we prove that $(2{\rm Var}\,[X(s)]\log\log({\rm Var}\,[X(s)]))^{-1/2}X(s)$ has the almost-sure cluster set $[-1,1]$ as $s\to 0+$. The latter result is applied to several coefficient sequences of number-theoretic origin.
