AI 中文总结
研究\(q = 4\)和\(q = 6\)时算术赫克三角形群\(\Gamma_q\)轨道的渐近密度，运用初等数论技术及相关函数性质与同余条件给出新证明，发现\(q = 4\)时互质整数对集合的划分特性。
AI 中文摘要
我们给出了新的证明来计算当\(q = 4\)和\(q = 6\)时算术赫克三角形群\(\Gamma_q\)轨道的渐近密度。我们使用初等数论技术以及莫比乌斯函数和黎曼ζ函数的基本性质，并附加同余条件。本笔记实际上源于观察到在\(q = 4\)的情况下，基础同余条件将互质整数对的集合划分为三个等密度的类。
英文摘要
We give new proofs computing the asymptotic densities for orbits of the arithmetic Hecke triangle groups $Γ_q$ when $q=4$ and $q=6$. We use elementary number theory techniques along with basic properties of the Möbius function and Riemann zeta function with additional congruence conditions. This note actually came about by observing that, in the $q=4$ case, the underlying congruence condition partitions the set of coprime integer pairs into three classes of equal density.
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