Let $\unicode[STIX]{x1D706}$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})=o(x)\end{eqnarray}$$ as $x\rightarrow \infty$, for any fixed natural numbers $a_{1},a_{2}$ and nonnegative integer $b_{1},b_{2}$ with $a_{1}b_{2}-a_{2}b_{1}\neq 0$. In this paper we establish the logarithmically averaged version $$\begin{eqnarray}\mathop{\sum }_{x/\unicode[STIX]{x1D714}(x)<n\leqslant x}\frac{\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})}{n}=o(\log \unicode[STIX]{x1D714}(x))\end{eqnarray}$$ of the Chowla conjecture as $x\rightarrow \infty$, where $1\leqslant \unicode[STIX]{x1D714}(x)\leqslant x$ is an arbitrary function of $x$ that goes to infinity as $x\rightarrow \infty$, thus breaking the ‘parity barrier’ for this problem. Our main tools are the multiplicativity of the Liouville function at small primes, a recent result of Matomäki, Radziwiłł, and the author on the averages of modulated multiplicative functions in short intervals, concentration of measure inequalities, the Hardy–Littlewood circle method combined with a restriction theorem for the primes, and a novel ‘entropy decrement argument’. Most of these ingredients are also available (in principle, at least) for the higher order correlations, with the main missing ingredient being the need to control short sums of multiplicative functions modulated by local nilsequences. Our arguments also extend to more general bounded multiplicative functions than the Liouville function $\unicode[STIX]{x1D706}$, leading to a logarithmically averaged version of the Elliott conjecture in the two-point case. In a subsequent paper we will use this version of the Elliott conjecture to affirmatively settle the Erdős discrepancy problem.

中文翻译：

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两点相关的对数平均 Chowla 和 ELLIOTT 猜想

让$\unicode[STIX]{x1D706}$表示刘维尔函数。在两点相关情况下，Chowla 猜想断言$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_ {2}n+b_{2})=o(x)\end{eqnarray}$$作为$x\rightarrow \infty$, 对于任何固定的自然数$a_{1},a_{2}$和非负整数$b_{1},b_{2}$和$a_{1}b_{2}-a_{2}b_{1}\neq 0$. 在本文中，我们建立了对数平均版本$$\begin{eqnarray}\mathop{\sum }_{x/\unicode[STIX]{x1D714}(x)<n\leqslant x}\frac{\unicode[STIX]{x1D706}(a_{1} n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})}{n}=o(\log \unicode[STIX]{x1D714}(x))\end {eqnarray}$$乔拉猜想为$x\rightarrow \infty$， 在哪里$1\leqslant \unicode[STIX]{x1D714}(x)\leqslant x$是一个任意函数$x$无穷大$x\rightarrow \infty$，从而打破了这个问题的“平价障碍”。我们的主要工具是Liouville 函数在小素数处的乘法性，这是Matomäki、Radziwiłł 和作者在短间隔内调制乘法函数平均值的最新结果、测度不等式的集中、Hardy-Littlewood 圆法结合a素数的限制定理，以及一个新颖的“熵减量论证”。大多数这些成分也可用于（至少原则上）高阶相关性，主要缺失的成分是需要控制由局部 nilsequences 调制的乘法函数的短和。我们的论点还扩展到比刘维尔函数更一般的有界乘法函数$\unicode[STIX]{x1D706}$，导致在两点情况下艾略特猜想的对数平均版本。在随后的论文中，我们将使用这个版本的艾略特猜想来肯定地解决 Erdős 差异问题。