Counting primes, groups, and manifolds.
Counting primes, groups, and manifolds.
复制标题
计算素数、群和流形。
DOI：
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发表时间：
2004
影响因子：
11.1
通讯作者：
L. Pyber
中科院分区：
文献类型：
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作者：
D. Goldfeld;A. Lubotzky;N. Nikolov;L. Pyber
设Lambda=SL(2)(Z)为模群，c(N)(Lambda)为至多为n的Lambda的同余子群的个数.我们证明了Lim(n--gt；无穷)(logc(N)(Lambda)/((Logn)(2)/logn))=(3-2(Sqrt)2)/4.证明是基于Bombieri-Vinogradov“平均Riemann假设”和组合数论中一类新的极值问题的解.对于高阶半单李群中格的子群增长，得到了类似的惊人的精确估计。如果G是这样一个李群，而Gamma是G的一个不可约格，则Gamma的子群增长与格无关，只依赖于G的直接因子的Lie类型，它可以很容易地从根系计算出来。这一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。这些证明使用了数论、代数群、有限群论和组合学的技巧。
Let Lambda=SL(2)(Z) be the modular group and let c(n)(Lambda) be the number of congruence subgroups of Lambda of index at most n. We prove that lim(n--> infinity )(log c(n)(Lambda)/((log n)(2)/log log n))=(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 Gamma is an irreducible lattice of G it turns out that the subgroup growth of Gamma 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.