Integers free of prime divisors from an interval. I, II. (Q2717618)

Let \(\Gamma(x,y,z)\) be the number of positive integers not exceeding \(x\) which are free of prime divisors from the interval \((z,y]\). Let \(\varrho\) and \(\omega\) denote the Dickman and Buchstab function respectively. Let \(u= \frac{\log x}{\log y}\) and \(v= \frac{\log x}{\log z}\) and define NEWLINE\[NEWLINE \eta(u,v):= \varrho(v)+\int_0^u \varrho (tv/u) \omega(u-t)\, dt, \quad (0< u \leq v).NEWLINE\]NEWLINE The author proves that uniformly for \(x \geq y \geq z \geq \frac{3}{2}\) one has \(\Gamma(x,y,z)= x \eta(u,v) + O(\frac{x}{\log y})\). He also derives a difference-differential equation for \(\eta(u,v)\).NEWLINENEWLINE In Part II the author relates the growth of \(\Gamma(x,y,z)\) with that of \(\Theta(x,y,z)\), which is defined to be the number of integers not exceeding \(x\) all of whose prime divisors are in \((z,y]\). The details are too technical to be stated here. The methods include complex integration together with the saddle point method. The results extend those of Buchstab, Tenenbaum, Friedlander, Saias and others.