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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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


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


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.
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  • 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, DOI 10.1016/S1385-7258(66)50029-4
  • P. Erdös, On some asymptotic formulas in the theory of the “factorisation numerorum.”, Ann. of Math. (2) 42 (1941), 989–993. MR 5516, DOI 10.2307/1968777
  • A. Oppenheim, 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).
  • 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
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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:
  • MathSciNet review: 682714