arXiv math/05113622005-11-14math.NT

关于偶Farey分数的一个密度定理

A density theorem on even Farey fractions

Cristian Cobeli, Alexandru Zaharescu

备注：: 35 pages, 6 figures

摘要：

令$F_Q$是$Q$阶的Farey序列，令$F_{Q,o}$和$F_{Q,e}$是分别具有奇数和偶数分解子的$Q$阶Farey分数的集合。$F_Q$的一个基本性质是任何一对相邻分数的乘积之和总是大于$Q$。对于$F_{Q,o}$和$F_{Q,e}$，此属性失败。之前计算了归一化对$(q'/Q,q''/Q)$的局部密度（如$Q\to\infty$），其中$q',q''$是$F_{Q,o}$中连续分数的乘积。密度在一系列四边形阶梯上以调和级数向点$(1,1)$递增。数值计算的小值的$Q$表明，这样的结果应该发生在偶数的情况下，而在奇数的情况下，相应的点的分布似乎更均匀。本文通过数值实验证明了，作为$Q\to\infty$，奇、偶情况下的局域密度是一致的。

英文摘要：

Let $F_Q$ be the Farey sequence of order $Q$ and let $F_{Q,o}$ and $F_{Q,e}$ be the set of those Farey fractions of order $Q$ with odd, respectively even denominators. A fundamental property of $F_Q$ says that the sum of denominators of any pair of neighbor fractions is always greater than $Q$. This property fails for $F_{Q,o}$ and for $F_{Q,e}$. The local density, as $Q\to\infty$, of the normalized pairs $(q'/Q,q''/Q)$, where $q',q''$ are denominators of consecutive fractions in $F_{Q,o}$, was computed previously. The density increases over a series of quadrilateral steps ascending in a harmonic series towards the point $(1,1)$. Numerical computations for small values of $Q$ suggest that such a result should rather occur in the even case, while in the odd case the distribution of the corresponding points appears to be more uniform. Reconciling with the numerical experiments, in this paper we show that, as $Q\to\infty$, the local densities in the odd and even case coincide.

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摘要：

英文摘要：

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