AI 中文总结
研究通过在特定弧内随机点评估归一化莫比乌斯傅里叶多项式\(P_N\)，给出黎曼假设的局部概率表述，证明其等价于随机变量局部矩的次多项式增长，用局部矩到点值不等式提供临界尺度局部准则。
AI 中文摘要
设\(P_N(t)=\frac{1}{\sqrt{N}}\sum_{n\leq N}\mu(n)e^{2\pi int}\)，\(t\in\mathbb{T}=\mathbb{R}/\mathbb{Z}\)。我们通过在半径为\(c/N\)的弧内的均匀随机点处评估归一化的莫比乌斯傅里叶多项式\(P_N\)，给出了黎曼假设的局部概率重新表述。我们证明黎曼假设等价于这些随机变量的任意高阶有限局部矩的次多项式增长。主要定量工具是一个局部矩到点值不等式，它从局部\(L^q\)数据中恢复值\(P_N(0)=M(N)/\sqrt{N}\)（其中\(M\)表示默滕斯函数）。这为黎曼假设提供了一个临界尺度局部准则，并补充了丹茹瓦的随机游走启发法、卡哈内的随机傅里叶级数理论以及莫比乌斯多项式的全局\(L^p\)半平坦性准则。
英文摘要
This article is primarily expository, but it also contains several new observations and reformulations concerning the Möbius function, probabilistic models, dynamical systems, and the Riemann hypothesis. Its starting question is classical: in what sense can the Möbius function be said to behave randomly? We begin with Denjoy's random-walk heuristic and place it in the context of later work on random multiplicative functions, short intervals, and Möbius pseudorandomness. We then develop two smoothing forms of the classical Mertens criterion for the Riemann hypothesis. The first uses the discrete Laplace transform \[ Φ(t)=\sum_{n\geq1}μ(n)e^{-nt} \] and identifies RH with the condition \[ Φ\in L^p(0,\infty) \qquad\text{for every }1\leq p<2. \] The second uses normalized Möbius Fourier polynomials and local moments on arcs of length comparable with \(1/N\). The paper also revisits the author's earlier criterion for RH in terms of discrete measures, as presented in Broughan's account of analytic equivalents of RH. Denjoy's heuristic is made precise in a simple independent coefficient model, and this model is carefully distinguished from the modern theory of random multiplicative functions. The resulting coefficient space also gives a measure--category contrast: the relevant \(L^p\)-property has full measure but is topologically meagre. A further point of the paper is dynamical: the multiplicative semigroup of positive integers acts naturally on the coefficient space, and the Möbius sequence is a distinguished arithmetic point for this action. The aim throughout is explanatory, while keeping these new observations visible: to show how arithmetic, probability, Fourier analysis, Mellin transforms, and dynamics fit together, and to indicate what modern results add to the older random-walk picture.
CommentsExtends and subsumes results of arXiv:2607.25002
