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

DOI:
https://doi.org/10.1090/S0002-9947-1983-0682714-6

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 $f’(0) = 0$ 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.

- R. R. Bahadur,
*Some limit theorems in statistics*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1971. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 4. MR**0315820** - N. G. de Bruijn,
*On the number of positive integers $\leq x$ and free prime factors $>y$. II*, Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math.**28**(1966), 239–247. MR**0205945** - P. Erdös,
*On some asymptotic formulas in the theory of the “factorisation numerorum.”*, Ann. of Math. (2)**42**(1941), 989–993. MR**5516**, DOI https://doi.org/10.2307/1968777
A. Oppenheim, - V. M. Zolotarev,
*On the closeness of the distributions of two sums of independent random variables*, Teor. Verojatnost. i Primenen.**10**(1965), 519–526 (Russian, with English summary). MR**0189109**

*On an arithmetic function*. II, J. London Math. Soc.

**2**(1927), 123-130. G. Szekeres and P. Turán,

*Über das zweite Hauptproblem der "Factorisatio Numerorum"*, Acta Litt. Sci. Szeged

**6**(1933), 143-154. J. Vaaler,

*The Berry-Esseen inequality and the central limit theorem*(to appear).

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Article copyright:
© Copyright 1983
American Mathematical Society