arXiv 1503.001992015-03math.NT

Farey分数的乘积

Products of Farey Fractions

Jeffery Lagarias, Harsh Mehta

备注：: 32 pages, 10 figures
发表信息：: Experimental Mathematics 26, No. 1, 1--21 (2017)
DOI：: 10.1080/10586458.2015.1020578

摘要：

$n$阶的{Farey分数} $F_n$由落在闭单位区间中的所有最低项的分数$\frac{h}{k}$组成，并且分母至多为$n$。本文考虑了所有非零的$n$阶Farey分数的乘积$F_n$.它研究了它们的增长$\log(F_n)$和它们的整除性能的固定素数，由$ord_p(F_n)$，作为$n$的功能的权力。$\log(F_n)$的增长与黎曼假设有关。本文从理论和经验两方面研究了函数$ord_p(F_n)$，给出了它们可能具有的几个未证明的性质（P1）~（P4）.它提出的证据提高的可能性，黎曼假设也可能是编码在$ord_p(F_n)$为一个单一的素数$p$。这种编码利用了这些产品的产品$G_n$的所有减少和未减少的法里分数的顺序$n$，这是连接莫比乌斯反演的关系。它涉及到新的算术函数混合的默比乌斯函数与功能的基数扩展到一个固定的总理基地$p$。

英文摘要：

The {Farey fractions} $F_n$ of order $n$ consist of all fractions $\frac{h}{k}$ in lowest terms lying in the closed unit interval and having denominator at most $n$. This paper considers the products $F_n$ of all nonzero Farey fractions of order $n$. It studies their growth measured by $\log(F_n)$ and their divisibility properties by powers of a fixed prime, given by $ord_p(F_n)$, as a function of $n$. The growth of $\log(F_n)$ is related to the Riemann hypothesis. This paper theoretically and empirically studies the functions $ord_p(F_n)$ and formulates several unproved properties (P1)-(P4) they may have. It presents evidence raising the possibility that the Riemann hypothesis may also be encoded in $ord_p(F_n)$ for a single prime $p$. This encoding makes use of a relation of these products to the products $G_n$ of all reduced and unreduced Farey fractions of order $n$, which are connected by Möbius inversion. It involves new arithmetic functions which mix the Möbius function with functions of radix expansions to a fixed prime base $p$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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