On a class of differential-difference equations arising in number theory
type: Item
instance of: scholarly article
title: On a class of differential-difference equations arising in number theory
zbMATH Open document ID: 0797.11072
DOI: 10.1007/BF02788841
published in: Journal d'Analyse Mathématique
publication date: 1994-10-20T00:00:00Z
review text: Functions defined by a differential-difference equation of the type (1) \(uf ' (u) + af(u) + bf(u-1) = 0\), where \(a\) and \(b\) are constants, arise frequently in number theory. Probably the best known examples are the Dickman function \(\rho(u)\) and the Buchstab function \(\omega (u)\). Functions satisfying (1) when \(a+b\) is an integer arise in sieve theory and in this context have been investigated by various authors. The equation (1) with general coefficients \(a\) and \(b\) have been studied by \textit{H. Iwaniec} [Acta Arith. 36, 171-202 (1980; Zbl 0435.10029)] and \textit{F. Wheeler} [Trans. Am. Math. Soc. 318, 491-523 (1990; Zbl 0697.10035)]. The object of the present paper is to describe, for any given pair of complex coefficients \((a,b)\) with \(b \neq 0\) the structure and asymptotic behavior of the general solution to \((1)\). Equation (1) with the initial condition (2) \(f(u) = \varphi(u)\) \((u_ 0 - 1 \leq u \leq u_ 0)\), where \(\varphi (u)\) is any given continuous function on \([u_ 0 - 1,u_ 0]\), has a unique continuous solution \(f(u) = f(u;\varphi)\) for \(u \geq u_ 0\). In the paper, the authors construct a set of ``fundamental'' solutions \(F(u)\) and \(F_ n(u)\) \((n \in \mathbb{Z})\) and the solution \(f(u)\) can be expressed as a convergent series \(f(u) = \alpha F(u) + \sum_{n \in \mathbb{Z}} \alpha_ n F_ n (u)\) with suitable coefficients \(\alpha\) and \(\alpha_ n\). The functions \(F\) and \(F_ n\) are defined by means of a contour integral, which can be estimated rather precisely. To state the result we set \[ \Phi (u,s) = {\exp \{-us + bI(s)\}s^{a+b-1} \over \sqrt {2 \pi u(1-1/s)}}, \] where \(I(s) = \int^ s_ 0{e^ z-1 \over z} dz\), and then we have the following result: For any fixed non-zero integer \(n\) and \(u \geq u_ 0 (\varepsilon,n)\) we have \[ F_ n(u) = (1+O({1 \over u})) \Phi (u, \zeta_ n) \] where \(\zeta_ n = \zeta_ n (u/b)\) is a certain complex solution of the equation \(e^ \zeta = 1 + {u \over b} \zeta\), and the im…
reviewed by: Xuan Tizuo
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 34K99
zbMATH DE Number: 528673
zbMATH Keywords: adjoint equation
zbMATH Keywords: differential-difference equation
zbMATH Keywords: asymptotic behavior
author: Adolf Hildebrand
author: Gérald Tenenbaum
MaRDI profile type: Publication
cites work: An Erdös-Kac theorem for integers without large prime factors
cites work: The general sieve
cites work: Q5520130
cites work: Differential-difference equations
cites work: Q5797342
cites work: Q5795180
cites work: Q5803354
cites work: Q5805143
cites work: Q5824821
cites work: Q3219268
cites work: A Differential Delay Equation Arising from the Sieve of Eratosthenes
cites work: Q3204561
cites work: Oscillation Theorems for Primes in Arithmetic Progressions and for Sifting Functions
cites work: NOTES ON FUNCTIONS CONNECTED WITH THE SIEVE
cites work: The Convolution Powers of the Dickman Function
cites work: The Asymptotic Behavior of the Solutions of a Class of Differential-Difference Equations
cites work: Rosser's sieve
cites work: An improvement of Selberg's sieve method I
cites work: Primes in short intervals
cites work: Two Differential-Difference Equations Arising in Number Theory
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description: scientific article; zbMATH DE number 528673