On difference-differential equations in the theory of sieves
type: Item
instance of: scholarly article
title: On difference-differential equations in the theory of sieves
zbMATH Open document ID: 0603.10046
DOI: 10.1016/0022-314X(86)90099-5
author: F. Grupp
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: 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. \(\kappa =1\), \(\alpha =3\), \(F_ 0(u)=2e^{\gamma}/u\), \(f_ 0(u)=0\).
Mathematics Subject Classification ID: 11N35
Mathematics Subject Classification ID: 34K10
zbMATH DE Number: 3972982
zbMATH Keywords: sieve methods
zbMATH Keywords: representations of functions
zbMATH Keywords: differential-difference equations
reviewed by: Sidney W. Graham
MaRDI profile type: Publication
cites work: The general sieve
cites work: A weighted sieve of Brun's type
cites work: The functions of the linear sieve
cites work: Q4052222
cites work: Rosser's sieve
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cites work: An improvement of Selberg's sieve method I
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description: scientific article; zbMATH DE number 3972982