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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the average number of groups of square-free order
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by Carl Pomerance PDF
Proc. Amer. Math. Soc. 99 (1987), 223-231 Request permission

Abstract:

Let $G(n)$ denote the number of (nonisomorphic) groups of order $n$. It is shown here that for large $x$ \[ x^{1.68} \leq \sum \nolimits ’_{n \leq x} G(n) \leq {x^2} \cdot \exp \{ -(1 + \mathrm {o}(1)) \log x\log \log \log x/\log \log x\} ,\] 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
  • © Copyright 1987 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 99 (1987), 223-231
  • MSC: Primary 11N45; Secondary 11N56, 20D60
  • DOI: https://doi.org/10.1090/S0002-9939-1987-0870776-1
  • MathSciNet review: 870776