Abc Allows Us to Count Squarefrees
نویسندگان
چکیده
We show several consequences of the abc-conjecture for questions in analytic number theory which were of interest to Paul Erd} os: For any given polynomial f(x) 2 Zx], we deduce, from the abc-conjecture, an asymptotic estimate for the frequency with which f(n) is squarefree, when n is an integer (and also deduce such estimates for binary homogenous forms). Amongst several applications of this result, we deduce that there is a squarefree number in every interval of length O(x ") around x, and give the asymptotic formula, predicted by Erd} os, for the average moments for the gaps between squarefree numbers.
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