Products of Ratios of Consecutive Integers |
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Authors: | Régis De La Bretèche Carl Pomerance Gérald Tenenbaum |
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Affiliation: | (1) École Normale Supérieure, Mathématiques et Applications, 45, rue d’ulm, 75230 Paris Cedex 05, France;(2) Department of Mathematics, Dartmouth College, Hanover, New Hampshire, 03755;(3) Institut Éĺie Cartan, Université Henri Poincaré Nancy 1, BP 239, 54506 Vandœuvre Cedex, France |
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Abstract: | Take the product of the numbers (n/(n+1))∊n for 1≤ n < N, where each ∊n is ± 1. Express the product as a/b in lowest terms. Evidently the minimal possible value for a over all choices for ∊n is 1; just take each ∊n = 1, or each ∊n = 0. Denote the maximal possible value of a by A(N). It is known from work of Nicolas and Langevin that (log 4+o(1))N≤ log A(N)≤(2/3+o(1))Nlog N. Using the Rosse–Iwaniec sieve, we improve the lower bound to the same order of magnitude as the upper bound.For Jean-Louis Nicolas, on his sixtieth birthday2000 Mathematics Subject Classification: Primary—11N56; Secondary—11N36 |
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Keywords: | extremal problems in number theory friable integers sieve largest prime factor |
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