On difference-differential equations in the theory of sieves
Many important functions in the theory of sieves may be viewed as solutions of the differential-difference equations \[ (u^{\kappa}F(u))'=\kappa u^{\kappa -1} f(u-1)\quad (u>\alpha),\quad (u^{\kappa}f(u))'=\kappa u^{\kappa -1} F(u-1)\quad (u>\beta), \] \[ F(u)=F_ 0(u)\;(0<u\leq \alpha),\quad f(u)=f_ 0(u)\;(0<u\leq \beta), \] where \(\kappa\), \(\alpha\), and \(\beta\) are constants and \(F_ 0\), \(f_ 0\) are given continuous functions. In this paper, representations of \(f\) and \(F\) are given that allow for easy calculation of these functions. The results generalize those of \textit{H.-E. Richert} and the author [J. Number Theory 22, 208--239 (1986; Zbl 0578.10048)], who considered the case that arises in the linear sieve; i.e. \(\kappa =1\), \(\alpha =3\), \(F_ 0(u)=2e^{\gamma}/u\), \(f_ 0(u)=0\).
scientific article; zbMATH DE number 3972982