论文标题
$ \ mathbb {z} _n $上的dirichlet字符和类似peisert的图形的超几何函数
Hypergeometric functions for Dirichlet characters and Peisert-like graphs on $\mathbb{Z}_n$
论文作者
论文摘要
对于Prime $ p \ equiv 3 \ pmod 4 $和一个正整数$ t $，令$ q = p^{2t} $。订单$ q $的PEISERT图是带顶点套装$ \ mathbb {f} _q $的图形，使得$ ab $如果$ a-b \ in \ langle g^4 \ rangle \ rangle \ cup g \ langle g^4 \ langle g^4 \ rangle $，其中$ g $是$ g $ of $ g $，而其中$ g $是$ \ mathbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb =在本文中，我们构建了一个带顶点集的类似图表，作为合适的$ n $的交换戒指$ \ mathbb {z} _n $，我们称之为\ textit {peisert {peisert like}图形，并用$ g^\ ast（n）$表示。由于需要$ \ mathbb {z} _n $的单位循环性，因此我们考虑$ n = p^α$或$ 2p^α$，其中$ p \ equiv 1 \ equiv 1 \ pmod 4 $是prime，$ $α$是一个正整数。对于Primes $ p \ equiv 1 \ pmod 8 $，我们通过评估某些字符总和来计算图$ g^\ ast（p^α）$中的三角形数量。接下来，我们研究$ g^\ ast（p^α）$的第4个订单集团。要查找$ g^\ ast（p^α）$中的订单$ 4 $的数量，我们首先引入包含dirichlet字符作为参数的超几何功能，然后以这些超眼函数的方式表示$ 4 $ in $ g^\ ast（p^α）$的订单$ 4 $的数量。
For a prime $p\equiv 3\pmod 4$ and a positive integer $t$, let $q=p^{2t}$. The Peisert graph of order $q$ is the graph with vertex set $\mathbb{F}_q$ such that $ab$ is an edge if $a-b\in\langle g^4\rangle\cup g\langle g^4\rangle$, where $g$ is a primitive element of $\mathbb{F}_q$. In this paper, we construct a similar graph with vertex set as the commutative ring $\mathbb{Z}_n$ for suitable $n$, which we call \textit{Peisert-like} graph and denote by $G^\ast(n)$. Owing to the need for cyclicity of the group of units of $\mathbb{Z}_n$, we consider $n=p^α$ or $2p^α$, where $p\equiv 1\pmod 4$ is a prime and $α$ is a positive integer. For primes $p\equiv 1\pmod 8$, we compute the number of triangles in the graph $G^\ast(p^α)$ by evaluating certain character sums. Next, we study cliques of order 4 in $G^\ast(p^α)$. To find the number of cliques of order $4$ in $G^\ast(p^α)$, we first introduce hypergeometric functions containing Dirichlet characters as arguments, and then express the number of cliques of order $4$ in $G^\ast(p^α)$ in terms of these hypergeometric functions.
