Localized factorizations of integers
نویسندگان
چکیده
منابع مشابه
Localized Factorizations of Integers
We determine the order of magnitude of H(k+1)(x,y, 2y), the number of integers n ≤ x that are divisible by a product d1 · · · dk with yi < di ≤ 2yi, when the numbers log y1, . . . , log yk have the same order of magnitude and k ≥ 2. This generalizes a result by Kevin Ford when k = 1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear i...
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We study the number of ways of writing a positive integer n as a product of integer factors greater than one. We survey methods from the literature for enumerating and also generating lists of such factorizations for a given number n. In addition, we consider the same questions with respect to factorizations that satisfy constraints, such as having all factors distinct. We implement all these m...
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Let D be the ring of integers in a finite extension of the rationals. The classic examination of the factorization properties of algebraic integers usually begins with the study of norms. In this paper, we show using the ideal class group, C(D), of D that a deeper examination of such properties is possible. Using the class group, we construct an object known as a block monoid, which allows us t...
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Write an integer as finite products of ordered factors belonging to a given subset P of integers larger than one. A very general central limit theorem is derived for the number of ordered factors in random factorizations for any subset P containing at least two elements. The method of proof is very simple and relies in part on Delange’s Tauberian theorems and an interesting Tauberian technique ...
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We study in detail the asymptotic behavior of the number of ordered factorizations with a given number of factors. Asymptotic formulae are derived for almost all possible values of interest. In particular, the distribution of the number of factors is asymptotically normal. Also we improve the error term in Kalmár’s problem of “factorisatio numerorum” and investigate the average number of distin...
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ژورنال
عنوان ژورنال: Proceedings of the London Mathematical Society
سال: 2010
ISSN: 0024-6115
DOI: 10.1112/plms/pdp056