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Transactions of the American Mathematical Society

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The number of factorizations of numbers less than $ x$ into factors less than $ y$


Author: Douglas Hensley
Journal: Trans. Amer. Math. Soc. 275 (1983), 477-496
MSC: Primary 10H25; Secondary 10K20, 60F10
MathSciNet review: 682714
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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^{\prime}(0) = 0$ such that if $ r = \sqrt {\log x} /\log y$ then as $ x \to \infty $ with $ r$ fixed,

$\displaystyle K(x,y) = x \exp\,\left({f(r)\,\sqrt {\log x} + O\,{{(\log \log x)}^2}} \right)$

. 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 [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0682714-6
Article copyright: © Copyright 1983 American Mathematical Society