论文标题
关于多项式论证的liouville功能
On the Liouville function at polynomial arguments
论文作者
论文摘要
令$λ$表示Liouville功能。 Chowla和Cassaigne-ferenczi-mauduit-rivat-sárközy提出的一个问题要求表明，如果$ p（x）\ in \ mathbb {z} [x] $，那么序列$λ（p（n））$ coply of n n $ p（x）$不是另一个py nem py nom py nom py nopy sartem sartem sartem sartem sartem sartem sartem sartem s propers n n n of py n n of py n n n of the p（p（n））$无限地变化。我们表明，如果（i）$ p $将序列$λ（p（n））$确实经常变化，则（i）$ p $分解为理由的线性因素；或（ii）$ p $是可还原的立方多项式；或（iii）$ p $将某种类型的任何数量二次化的产物分解为某种类型的乘积；或（iv）$ p $是不属于特殊密度零集的任何多项式。关于（i），我们更普遍地证明了$ g（p（n））$ $ g $的部分总和在$ g $的必要和足够条件下表现出非平凡的取消。这建立了埃利奥特（Elliott）的猜想的“ 99％版本”，用于以某种顺序的统一性为基础。 （iv）部分还概括为$ g（p（n））$的设置，并提供了Skorobogatov和沙发的最新结果的乘法函数类似物，几乎所有的多项式都达到了素数。
Let $λ$ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-Sárközy asks to show that if $P(x)\in \mathbb{Z}[x]$, then the sequence $λ(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial. We show that the sequence $λ(P(n))$ indeed changes sign infinitely often, provided that either (i) $P$ factorizes into linear factors over the rationals; or (ii) $P$ is a reducible cubic polynomial; or (iii) $P$ factorizes into a product of any number of quadratics of a certain type; or (iv) $P$ is any polynomial not belonging to an exceptional set of density zero. Concerning (i), we prove more generally that the partial sums of $g(P(n))$ for $g$ a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on $g$. This establishes a "99% version" of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of $g(P(n))$ and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value.
