Odd order cases of the logarithmically averaged Chowla conjecture
Odd order cases of the logarithmically averaged Chowla conjecture
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对数平均 Chowla 猜想的奇数阶情况
DOI：
10.5802/jtnb.1062
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发表时间：
2017
影响因子：
0.4
通讯作者：
Joni Teravainen
中科院分区：
文献类型：
--
作者：
T. Tao;Joni Teravainen
Chowla的一个著名猜想指出，Liouville函数$\lambda(n)$与其移位的相关性可以忽略不计。最近，作者建立了关于相乘函数相关性的对数平均Elliott猜想的一个弱形式，从而推导出对数平均Chowla猜想的所有奇阶情况。本文给出了对数平均Chowla猜想的奇阶情形的一个新的较短的证明。特别是，这个证明完全避免了对遍历论的提及，而遍历论在前面的证明中起着重要的作用。
A famous conjecture of Chowla states that the Liouville function $\lambda(n)$ has negligible correlations with its shifts. Recently, the authors established a weak form of the logarithmically averaged Elliott conjecture on correlations of multiplicative functions, which in turn implied all the odd order cases of the logarithmically averaged Chowla conjecture. In this note, we give a new and shorter proof of the odd order cases of the logarithmically averaged Chowla conjecture. In particular, this proof avoids all mention of ergodic theory, which had an important role in the previous proof.