On the distribution in short intervals of integers having no large prime factor (Q1084440)

Let \(\Psi\) (x,y) be the number of integers \(\leq x\) all of whose prime factors are \(\leq y\). The authors investigate the function \(\Psi (x, x^{\alpha})-\Psi (x-x^{\beta}, x^{\alpha})\) with particular regard to when it is greater than zero for large x and when it is greater than \(c(\alpha,\beta)x^{\beta}\) for some positive constant c(\(\alpha\),\(\beta)\) for large x. Define f(\(\alpha)\) to be the infimum of the values of \(\beta\) for which, for all \(\alpha_ 1>\alpha\), it is true that \(\Psi (x, x^{\alpha_ 1})-\Psi (x-x^{\beta}, x^{\alpha_ 1})>0\) for large x and define \(f^*(\alpha)\) similarly where this difference is greater than \(c(\alpha_ 1,\beta)x^{\beta}\). The authors show that there exists a positive constant c such that \(f^*(\alpha)\leq 1-\alpha -c\alpha (1-\alpha)^ 3\) for \(0\leq \alpha \leq 1\), and that \(f(\alpha)\leq 1-2\alpha (1-2^{-[\alpha^{-1}]})\) for \(0<\alpha \leq 1/2\). They also show that for fixed \(\alpha\) and \(\beta\) and almost all x, \[ \Psi (x, x^{\alpha})-\Psi (x-x^{\beta}, x^{\alpha})\geq (1/64)\quad \beta \rho (1/\alpha) x^{\beta}, \] where \(\rho\) (t) is Dickman's function.