期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2004
卷号：101
期号：37
页码：13428-13430
DOI：10.1073/pnas.0404571101
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：Let [IMG]f1.gif" BORDER="0"> be the modular group and let cn({Lambda}) be the number of congruence subgroups of {Lambda} of index at most n. We prove that [IMG]f2.gif" BORDER="0">. 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.
