## On a class of differential-difference equations arising in number theory.(English)Zbl 0797.11072

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 H. Iwaniec [Acta Arith. 36, 171-202 (1980; Zbl 0435.10029)] and 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 implied constant depends at most on $$\varepsilon$$ and $$n$$.
The principal result of the paper is as follows. Let $$\varphi (u)$$ be a continuous function on $$[u_ 0 - 1,u_ 0]$$ and let $$f(u) = f(u;\varphi)$$ be the unique continuous solution to (1) and (2). Then we have $f(u) = \alpha F(u) + \sum_{\alpha \in \mathbb{Z}} \alpha_ n F_ n(u), \quad u>u_ 0+1 \tag{3}$ where $$\alpha = \langle \varphi, G \rangle$$, $$\alpha_ n = \langle \varphi, G_ n \rangle$$ and the series in (3) is uniformly convergent for $$u \geq u_ 0 + 1 + \delta$$, for any fixed $$\delta>0$$.
Moreover, the authors derive several corollaries from the principal result.

### MSC:

 11N25 Distribution of integers with specified multiplicative constraints 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)

### Citations:

Zbl 0435.10029; Zbl 0697.10035
Full Text:

### References:

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