Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

  • [1] R. R. Bahadur, Some limit theorems in statistics, Chap. 2, SIAM, Philadelphia, Pa., 1971. MR 0315820 (47:4369)
  • [2] N. G. deBruijn, On the number of positive integers $ \leqslant x$ and free of prime factors $ > y$. II, Nederl. Akad. Wetensch. Proc. Ser. A 69 (1966), 239-247. MR 0205945 (34:5770)
  • [3] P. Erdös, On some asymptotic formulas in the theory of "factirsatio numerorum", Ann. of Math. (2) 42 (1941), 989-993. Corrections to two of my papers, Ann. of Math. (2) 44 (1943), 647-651. MR 0005516 (3:165b)
  • [4] A. Oppenheim, On an arithmetic function. II, J. London Math. Soc. 2 (1927), 123-130.
  • [5] G. Szekeres and P. Turán, Über das zweite Hauptproblem der "Factorisatio Numerorum", Acta Litt. Sci. Szeged 6 (1933), 143-154.
  • [6] J. Vaaler, The Berry-Esseen inequality and the central limit theorem (to appear).
  • [7] V. M. Zolotarev, On the closeness of the distributions of two sums of independent random variables, Theory Probab. Appl. 10 (1965), 472-479. MR 0189109 (32:6536)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10H25, 10K20, 60F10

Retrieve articles in all journals with MSC: 10H25, 10K20, 60F10

Additional Information

Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society