科研之友
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摘要
For k1 an integer and x1 a real number, let k(x) be the number of integers smaller than x having exactly k distinct prime divisors. Building on recent work of Matomaki and Radziwi, we investigate the asymptotic behavior of k(x+h)-k(x) for almost all x, when h is very small. We obtain optimal results for klog2x and close to optimal results for 5klog2x. Our method also applies to y-friable integers in almost all intervals [x,x+h] when (logx/logy)(logx)1/6-epsilon.
