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Integers free of large prime divisors in short intervals. (English) Zbl 0562.10018

Let \(\psi\) (x,y) denote the number of positive integers \(\leq x\) composed only of prime factors \(\leq y\). The asymptotic behaviour of \(\psi\) (x,y) as x,y\(\to \infty\) is quite well understood but not much is known about the difference \(\psi (x+z,y)-\psi (x,y)\). In this paper the author derives a variety of upper bounds for this difference, these bounds depending on the relative sizes of x,y and z. Some applications are also considered. A particularly striking result is the ”sub-additivity of \(\psi\) ”, namely that \(\psi (x+z,y)\leq \psi (x,y)+\psi (z,y)\) holds for all \(x,z\geq y\geq y_ 0\) (fixed). The methods used are elementary and again depend in an essential way on the identity \[ \psi (x,y) \log x- \int^{x}_{1}\psi (t,y)dt/t=\sum_{p^ m\leq x, p\leq y}\psi (x/p^ m,y) \log p \] which the author has used in some of his earlier papers quite successfully.
Reviewer: K.Alladi

MSC:

11N05 Distribution of primes
11N37 Asymptotic results on arithmetic functions
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