发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了埃尔德什问题#522，证明随机Littlewood多项式的单位圆盘零点比例收敛至1/2，建立了随机幂级数零点的通用径向定律，其波动依赖系数律的四阶累积量，且结果已在Lean 4中形式验证。
AI 中文摘要
我们解决了1961年提出的埃尔德什问题#522，证明对于随机Littlewood多项式，闭单位圆盘内零点的比例几乎必然收敛至1/2。更一般地，对于具有独立同分布、有界、非退化且中心对称的实高斯或圆复高斯系数的随机幂级数的部分和，我们证明半径为1+x/n的圆盘内零点的比例几乎必然收敛至(1/2)(1 + coth x - 1/x)，该收敛在x∈ℝ上一致成立。因此几乎所有零点都位于距单位圆盘O(1/n)的距离处。对于随机Littlewood多项式，即使次数间存在任意依赖，单位圆盘处的收敛也以几乎必然速率O(n^{-1/4}√log n)成立。我们建立了依赖序列的对数迭代律准则，对于实高斯系数，在所有整数次数、方差及跨次数协方差渐近、定量中心极限定理、联合高斯极限和增量边界上得到了精确的对数迭代律。对于具有有界密度和所有有限矩的对称系数律，闭单位圆盘内零点的方差为(c_G + κ₄/12)n + O_ξ(n^{399/400})，对于随机符号，该常数为c_G - 1/6。因此径向定律是通用的，而波动则通过其四阶累积量κ₄依赖于系数律。我们的所有主要结果均在Lean 4中得到形式验证，且我们对埃尔德什问题#522的证明已被接受为Google DeepMind的形式化猜想项目中其形式化表述的解决方案。
英文摘要
We resolve Erdős Problem #522, posed in 1961, by proving that for random Littlewood polynomials the proportion of zeros in the closed unit disk converges almost surely to $1/2$. More generally, for partial sums of random power series with i.i.d. coefficients that are bounded, nondegenerate, and centrally symmetric, real Gaussian, or circular complex Gaussian, we prove that the proportion of zeros in the disk of radius $1+x/n$ converges almost surely to $\frac12(1+\coth x-1/x)$, uniformly in $x\in\mathbb{R}$. Thus almost all zeros lie at distance $O(1/n)$ from the unit circle. For random Littlewood polynomials, even with arbitrary dependence between degrees, the convergence at the unit circle holds at the almost-sure rate $O(n^{-1/4}\sqrt{\log n})$. We establish a law-of-the-iterated-logarithm criterion for dependent sequences and, for real Gaussian coefficients, obtain the sharp law of the iterated logarithm over all integer degrees, variance and cross-degree covariance asymptotics, quantitative central limit theorems, joint Gaussian limits, and increment bounds. For symmetric coefficient laws with bounded density and all moments finite, the number of zeros in the closed unit disk has variance $(c_G+κ_4/12)n+O_ξ(n^{399/400})$, and for random signs the constant is $c_G-1/6$. Thus the radial law is universal, while the fluctuations depend on the coefficient law through its fourth cumulant $κ_4$. All our main results are formally verified in Lean 4, and our proof of Erdős Problem #522 has been accepted as a solution to its formal statement in Google DeepMind's Formal Conjectures project.
Comments82 pages, 4 figures, 2 tables. Lean 4 proofs of all main results and the data behind the figures and tables at https://github.com/chreia/erdos-522
