论文标题
平滑整数和de bruijn的近似$λ$
Smooth integers and de Bruijn's approximation $Λ$
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
This paper is concerned with the relationship of $y$-smooth integers and de Bruijn's approximation $Λ(x,y)$. Under the Riemann hypothesis, Saias proved that the count of $y$-smooth integers up to $x$, $Ψ(x,y)$, is asymptotic to $Λ(x,y)$ when $y \ge (\log x)^{2+\varepsilon}$. We extend the range to $y \ge (\log x)^{3/2+\varepsilon}$ by introducing a correction factor that takes into account the contributions of zeta zeros and prime powers. We use this correction term to uncover a lower order term in the asymptotics of $Ψ(x,y)/Λ(x,y)$. The term relates to the error term in the prime number theorem, and implies that large positive (resp. negative) values of $\sum_{n \le y} Λ(n)-y$ lead to large positive (resp. negative) values of $Ψ(x,y)-Λ(x,y)$, and vice versa. Under the Linear Independence hypothesis, we show a Chebyshev's bias in $Ψ(x,y)-Λ(x,y)$.
