计数素数、群和流形
Counting primes, groups and manifolds
摘要：
设$Λ= \mathrm{SL}_2(\Bbb Z)$是模群，$c_n(Λ)$是指数至多为$n$的$Λ$的同余子群的个数。本文证明了$\lim\limits_{n\to \infty} \frac{\log c_n(Λ)}{(\log n)^2/\log\log n} = \frac{3-2\sqrt{2}}{4}.$这个证明是基于Bjueri-Vinogradov的“平均上的黎曼假设”和组合数论中一个新的极值问题的解.对于高阶半单李群中格的子群增长也得到了类似的惊人的精确估计。如果$G$是这样一个李群，而$Γ$是$G$的一个不可约格，则证明了$Γ$的子群增长与格无关，只取决于$G$的直接因子的Lie类型。它可以很容易地从根系计算。这个结果的最一般情况依赖于广义黎曼假设，但许多特殊情况是无条件的。证明使用的技术从数论，代数群，有限群论和组合。
英文摘要：
Let $Λ= \mathrm{SL}_2(\Bbb Z)$ be the modular group and let $c_n(Λ)$ be the number of congruence subgroups of $Λ$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log c_n(Λ)}{(\log n)^2/\log\log n} = \frac{3-2\sqrt{2}}{4}.$ The proof is based on the Bombieri-Vinogradov `Riemann hypothesis on the average' and on the solution of a new type of extremal problem in combinatorial number theory. Similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple Lie groups. If $G$ is such a Lie group and $Γ$ is an irreducible lattice of $G$ it turns out that the subgroup growth of $Γ$ is independent of the lattice and depends only on the Lie type of the direct factors of $G$. It can be calculated easily from the root system. The most general case of this result relies on the Generalized Riemann Hypothesis but many special cases are unconditional. The proofs use techniques from number theory, algebraic groups, finite group theory and combinatorics.
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