On difference-differential equations in the theory of sieves

Abstract

Let F, f be the continuous solutions of the following system of difference-differential equations: F( u) = F 0( u), 0 < u ≦ α; f( u) = f 0( u), 0 < u ≦ β; ( u κ F( u))′ = κu κ − 1 f( u − 1), ( u κ F( u))′ = κu κ − 1 F( u − 1), u > β; where κ, α, β are constants and F 0, f 0 are “initial” functions. Many important functions in number theory can be represented by F and f, if a special choice of α, β, κ, F 0, f 0 is made, for example, Selberg's function σ, Buchstab's function w, Dickman's function ϱ (for definitions see Halberstam and Richert (“Sieve Methods,” Academic Press, London, 1974)), the functions F, f of the κ-dimensional Rosser-sieve ( Iwaniec, Acta Arith. 36 (1980) , 171–202), the functions F, f of the κ-dimensional iterated Selberg-sieve ( Iwaniec, van de Lune, and de Riele, Indag. Math. 42 (1980) , 409–417). In this paper a new representation of F, f is given which allows, for example, an easy calculation of these functions. The case κ = 1, α = 3, β = 2, F 0(u) = 2e γ u , f 0( u) = 0 was treated in a paper of Richert and the author ( J. Number Theory, in press).