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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The number of factorizations of numbers less than $ x$ into divisors greater than $ y$


Author: Douglas Hensley
Journal: Trans. Amer. Math. Soc. 282 (1984), 259-274
MSC: Primary 11N25
MathSciNet review: 728712
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Abstract: Let $ A(x,\,y)$ be the number in the title. There is a function $ h:[0,\,\infty ) \to [0,\,2]$, decreasing and convex, with $ h(0) = 2$ and $ \operatorname{lim}_{r \to \infty }h(r) = 0$, such that if $ r = \operatorname{log} y/\sqrt {\operatorname{log} \,x} $ then as $ x \to \infty $ with $ r$ fixed,

$\displaystyle A(x,y) = \frac{{C(r)x\,\operatorname{exp} \left( {h(r)\sqrt {\ope... ...{log} \,x)}^{3/4}}}}\left( {1 + O{{(\operatorname{log} \,x)}^{ - 1/4}}} \right)$

. The estimate is uniform on intervals $ 0 < r \leq {R_0}$. As corollaries we have for $ \operatorname{log} \,y = \theta {(\operatorname{log} \,x)^{1/4}}$,

$\displaystyle \lim \limits_{x \to \infty } \,\frac{{A(x,\,y)}} {{A(x,\,1)/y}} = {e^{{\theta ^2}/2}}$

,and if $ \operatorname{log} \,y = o {(\operatorname{log} \,x)^{1/4}}$ then $ A(x,\,y) \approx A(x,\,1)/y$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0728712-6
PII: S 0002-9947(1984)0728712-6
Keywords: Complex analytic methods, elementary probability methods, large deviations
Article copyright: © Copyright 1984 American Mathematical Society