Let $\mu (g)$ be the largest possible order of a group of automorphisms of a compact Riemann surface of genus $g\ge 2$. The celebrated results of Accola-Maclachlan and Hurwitz-Macbeath, taken together say that $\mu (g)$ ranges between $8(g+1)$ and $84(g-1)$ and these two bounds are attained for infinitely many g. Here we prove that given a prime $q>167$ congruent to −1 modulo 3, not congruent +1 modulo 5 and a prime $p\ge 84q$, $\mu (g)=(8(q+4)/q)(g-1)$ for $g=q{p}^m+1$ for infinitely many integers m. Furthermore we also prove that for any prime q for which $p=q+2$ is a prime, $\mu (g)=(12(q+2)/q)(g-1)$ for $g=q{p}^m+1$ for infinitely many m. These mean that there exist an infinite rational sequence $(r_n)$ convergent to 8 and, assuming the truth of the Twin Prime Conjecture, an infinite rational sequence $(s_n)$ convergent to 12 together with infinite sets $R_n$ and $S_n$ of integers for which$\mu (g)=\{\begin{array}{ccc}{r}_n(g-1)\hfill & \text{if}\hfill & g\in R_n\hfill \\ s_n(g-1)\hfill & \text{if}\hfill & g\in S_n\hfill \end{array}$ and $8,12$ are the unique numbers for which this can happen. A consequence of these results is the full understanding of the asymptotic's of μ. Namely if instead of $\mu =\mu (g)$ we consider its normalization $\tilde{\mu }(g)=\mu (g)/g$ then $\{8\}\subseteq {({\mathcal{M}}^d)}^d\subseteq \{8,12\}$, where $\mathcal{M}$ stands for the set of values of $\tilde{\mu }$ and the operator d for the set of accumulation points and, assuming the truth of the Twin Prime Conjecture, $12\in {({\mathcal{M}}^d)}^d$.

让 $\mu （G）$ 是属的紧致黎曼曲面的一组自同构的最大可能阶 $G\ge \mathrm{2个}$。结合Accola-Maclachlan和Hurwitz-Macbeath的著名成绩，他们说：$\mu （G）$ 介于 $8（G+\mathrm{1个}）$ 和 $84（G-\mathrm{1个}）$并且这两个界限达到无限大的g。在这里我们证明给定素数$q>167$ 等于-1模3，不等于+1模5和素数 $p\ge 84q$， $\mu （G）=（8（q+4）/q）（G-\mathrm{1个}）$ 为了 $G=q{p}^{米}+\mathrm{1个}$对于无限多个整数m。此外，我们还证明了对任意素数q为这$p=q+\mathrm{2个}$ 是素数 $\mu （G）=（12（q+\mathrm{2个}）/q）（G-\mathrm{1个}）$ 为了 $G=q{p}^{米}+\mathrm{1个}$无限数m。这意味着存在无限的有理序列$（{\mathrm{[R}}_{ñ}）$ 收敛至8，并假设孪生素数猜想的真相，则一个无穷有理数列 $（s_{ñ}）$ 收敛到12与无限集 ${\mathrm{[R}}_{ñ}$ 和 ${\mathrm{小号}}_{ñ}$ 的整数$\mu （G）=\{\begin{array}{ccc}{\mathrm{[R}}_{ñ}（G-\mathrm{1个}）\hfill & \text{如果}\hfill & G\in {\mathrm{[R}}_{ñ}\hfill \\ s_{ñ}（G-\mathrm{1个}）\hfill & \text{如果}\hfill & G\in {\mathrm{小号}}_{ñ}\hfill \end{array}$ 和 $8，12$是可能发生的唯一编号。这些结果的结果是充分了解μ的渐近性。即如果不是$\mu =\mu （G）$ 我们考虑它的归一化 $\stackrel{〜}{\mu }（G）=\mu （G）/G$ 然后 $\{8\}\subseteq {（{\mathcal{中号}}^d）}^d\subseteq \{8，12\}$， 在哪里 $\mathcal{中号}$ 代表的一组值 $\stackrel{〜}{\mu }$算子d为一组累积点，并假设Twin Prime Conjecture的真相，$12\in {（{\mathcal{中号}}^d）}^d$。