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  ABC allows us to count squarefrees (1998) [7 citations — 0 self]

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by Andrew Granville
Internat. Math. Res. Notices
http://www.math.uga.edu/~andrew/Postscript/polysq3.ps
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Abstract:

Dedicated to the memory of Paul Erdos Abstract. We show several consequences of the abc-conjecture for questions in analytic number theory which were of interest to Paul Erdos: For any given polynomial f(x) 2 Z[x], 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 Erdos, for the average moments for the gaps between squarefree numbers. 1. Introduction. For any given polynomial f(x) 2 Z[x], we investigate what proportion of the

Citations

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