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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The number of factorizations of numbers less than $x$ into factors less than $y$
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by Douglas Hensley PDF
Trans. Amer. Math. Soc. 275 (1983), 477-496 Request permission

Abstract:

Let $K(x,y)$ be the number in the title. There is a function $f(r)$, concave and decreasing with $f(0) = 2$ and $f’(0) = 0$ such that if $r = \sqrt {\log x} /\log y$ then as $x \to \infty$ with $r$ fixed, \[ K(x,y) = x \exp \left ({f(r) \sqrt {\log x} + O {{(\log \log x)}^2}} \right )\]. The proof uses a uniform version of Chernoff’s theorem on large deviations from the sample mean of a sum of $N$ independent random variables.
References
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 275 (1983), 477-496
  • MSC: Primary 10H25; Secondary 10K20, 60F10
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0682714-6
  • MathSciNet review: 682714