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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the average number of groups of square-free order


Author: Carl Pomerance
Journal: Proc. Amer. Math. Soc. 99 (1987), 223-231
MSC: Primary 11N45; Secondary 11N56, 20D60
MathSciNet review: 870776
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Abstract: Let $ G(n)$ denote the number of (nonisomorphic) groups of order $ n$. It is shown here that for large $ x$

$\displaystyle {x^{1.68}} \leq \mathop {\sum '}\limits_{n \leq x} G(n) \leq {x^2... ...dot {\text{ exp\{ - (1 + }}o{\text{(1))}}\log x\log \log \log x/\log \log x\} ,$

where $ \sum '$ denotes a sum over square-free $ n$. Under an unproved hypothesis on the distribution of primes $ p$ with all primes in $ p - 1$ small, it is shown that the upper bound is tight.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0870776-1
PII: S 0002-9939(1987)0870776-1
Article copyright: © Copyright 1987 American Mathematical Society