论文标题
将整数的限制分区计算为分数：生成功能的对称和模式以及与$ω（t）$的连接
Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to $ω(t)$
论文作者
论文摘要
在研究整数分区的动机上，我们将整数分区分配为特定形式的分数，即具有恒定的分母和独特的奇数甚至分子。当分子是奇数时，整数的分区数小于分母形式形式的对称模式。如果条款数仅限于$ h $，则生成函数的非零项是单峰的，而整数$ h $具有最多的分区。此类特性可以应用于特定类别的非线性双苯胺方程。我们还使用分子检查分区。我们证明，有$ 2^{ω（t）} -2 $ $ t $分数的分数分数，第一个$ x $连续的分子甚至是分子的整数，而等于$ y $的同等分母，其中$ 0 <y <x <x <t $。然后，我们将其用于产生推论，例如Dirichlet系列身份，以及将质量欧米茄函数扩展到复杂平面，尽管该扩展并非到处都是分析。
Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to $h$, then the nonzero terms of the generating function are unimodal, with the integer $h$ having the most partitions. Such properties can be applied to a particular class of nonlinear Diophantine equations. We also examine partitions with even numerators. We prove that there are $2^{ω(t)}-2$ partitions of an integer $t$ into fractions with the first $x$ consecutive even integers for numerators and equal denominators of $y$, where $0<y<x<t$. We then use this to produce corollaries such as a Dirichlet series identity and an extension of the prime omega function to the complex plane, though this extension is not analytic everywhere.
