论文标题
在一类几乎纽曼多项式模型和密度定理上
On a class of lacunary almost Newman polynomials modulo p and density theorems
论文作者
论文摘要
$β$ -Shift的动力学Zeta函数$ζ_β（z）$ shift的减少模量$ p $与$β$ -Shift的动力学Zeta函数$β> 1 $相关，以$β> 1 $接近一个。我们简要回想起这个家庭与莱默的问题如何相关。通过Kronecker的平均值定理（被视为统一分布理论的经典定理的类似物），关于$ \ mathbb {f} _p $中的零数量及其因素化提出了各种问题。这些问题是使用Schinzel的结果部分回答的，该结果由Sawin，Shusterman和Stoll以及密度定理（Frobenius，Chebotarev，Serre，Rosen）重新审视。这些问题是由于寻找Mahler测度的整数多项式存在> 1比最小的塞勒姆编号1.176280少1。在$ \ Mathbb {f} _p $中，这些多项式的零数量的模块化形式（或模块化表示）的显式连接是在少数情况下获得的。总的来说，这是可以预期的，因为它必须根据兰兰兹计划存在。
The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $ζ_β(z)$ of the $β$-shift, for $β> 1$ close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in $\mathbb{F}_p$ and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in $\mathbb{F}_p$ is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.
