幻方、对称群与默比乌斯随机性
Magic squares, the symmetric group and Möbius randomness
AI总结：
本文利用特征映射给出了CUE系综长期系数矩的新组合证明，扩展至对称幂迹的矩，并提出了关于Liouville或Möbius函数扭曲特征和矩的猜想，在函数域中验证，且给出了矩阵积分的新公式。
摘要：
Diaconis和Gamburd计算了CUE系综中长期系数的矩。我们使用特征映射给出了其结果的一个新组合证明。我们还将他们的计算扩展到对称幂迹的矩，其中相同结果成立但范围更广。我们的组合证明受gcd矩阵启发，正如Vaughan和Wooley以及Granville和Soundararajan所使用的那样。我们利用这些CUE计算提出了关于由Liouville（或Möbius）函数扭曲的特征和矩的猜想，并在函数域中建立了其一个版本。我们猜想的寓意（及其在函数域中的验证）是，Steinhaus随机乘性函数是由随机Dirichlet特征扭曲的Liouville（或Möbius）函数的良好模型。我们还在不对矩阵大小作任何条件的情况下评估了长期系数和对称幂迹的矩。作为应用，我们给出了Keating、Rodgers、Roditty-Gershon和Rudnick在研究$k$重除数函数时考虑的矩阵积分的新公式。
英文摘要：
Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the $k$-fold divisor function.
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