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Counting primes, groups and manifolds

Abstract:

Let $\Lambda = \mathrm{SL}_2(\Bbb Z)$ 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 $\lim\limits_{n\to \infty} \frac{\log c_n(\Lambda)}{(\log n)^2/\log\log n} = \frac{3-2\sqrt{2}}{4}.$ 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 s...

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Publisher copy:
10.1073/pnas.0404571101

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Journal:
Proceedings of the National Academy of Sciences of the United States of America
Volume:
101
Issue:
37
Pages:
13428-13430
Publication date:
2004-06-09
DOI:
EISSN:
1091-6490
ISSN:
0027-8424
URN:
uuid:35add1d3-72f0-4301-b347-09f303c6cf16
Source identifiers:
354383
Local pid:
pubs:354383
Language:
English
Keywords:

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