论多项式参数下的柳维尔函数

IF 1.7 1区 数学 Q1 MATHEMATICS
Joni Teräväinen
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引用次数: 0

摘要

摘要:让$\lambda$表示Liouville函数。Chowla和Cassaigne--Ferenczi--Mauduit--Rivat--S'ark\"ozy提出的一个问题要求证明,如果$P(x)\in\mathbb{Z}[x]$,那么序列$\lambda(P(n))$会无限频繁地改变符号,前提是$P(x)$不是另一个多项式的平方。我们证明$\lambda(P(n))$ 序列确实会无限频繁地改变符号,前提是:(i) $P$ 分解为有理数上的线性因数;或 (ii) $P$ 是可还原的三次多项式;或 (iii) $P$ 分解为任意数量的某类二次项的乘积;或 (iv) $P$ 是不属于密度为零的特殊集合的任意多项式。关于第(i)项,我们更一般地证明,对于$g$ 一个有界乘法函数,在关于$g$ 的必要条件和充分条件下,$g(P(n))$ 的部分和表现出非对称取消。这确立了艾略特猜想的 "99% 版本",即乘法函数取值于某阶的同根。第(iv)部分还概括了$g(P(n))$ 的设置,并提供了斯科罗博加托夫和索福斯关于几乎所有达到素值的多项式的最新结果的乘法函数类比。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the Liouville function at polynomial arguments

abstract:

Let $\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne--Ferenczi--Mauduit--Rivat--S\'ark\"ozy asks to show that if $P(x)\in\mathbb{Z}[x]$, then the sequence $\lambda(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial.

We show that the sequence $\lambda(P(n))$ 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.

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来源期刊
CiteScore
3.20
自引率
0.00%
发文量
35
审稿时长
24 months
期刊介绍: The oldest mathematics journal in the Western Hemisphere in continuous publication, the American Journal of Mathematics ranks as one of the most respected and celebrated journals in its field. Published since 1878, the Journal has earned its reputation by presenting pioneering mathematical papers. It does not specialize, but instead publishes articles of broad appeal covering the major areas of contemporary mathematics. The American Journal of Mathematics is used as a basic reference work in academic libraries, both in the United States and abroad.
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