ABC allows us to count squarefrees
ABC allows us to count squarefrees
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ABC 允许我们计算 squarefree
DOI：
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
A. Granville
中科院分区：
文献类型：
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作者：
A. Granville
我们展示了Paul Erd} os感兴趣的解析数论问题的abc猜想的几个结果：对于任何给定的多项式f（x）2 Zx]，我们从abc猜想中推导出当n是整数时f（n）无平方的频率的渐近估计（也推导出二元齐次形式的估计）。在这个结果的几个应用中，我们推导出在x周围的每一个长度为O（x“）的区间中存在一个无平方数，并给出了Erd} os预言的无平方数间隔的平均矩的渐近公式.
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.