Sur les puissances de convolution de la fonction de Dickman. (On the convolution powers of the Dickman function). (French) Zbl 0881.11069

Smida, Hikma

Acta Arith. 59, No. 2, 123-143 (1991).

Let \(\tau_k(n)\) be the number of representations of \(n\) as a product of \(k\) factors and let \(P(n)\) be the greatest prime factor of \(n\). N. G. de Bruijn and J. H. van Lint [Ned. Akad. Wet., Proc., Ser. A 67, 339-359 (1964; Zbl 0131.28703){]} studied the expression \(\sum_{n\leq x,P(n)\leq y} \tau_k(n)\), showing that it is asymptotic to \(x\rho_k(x)(\log z)^{k-1}\) as \(x,y\to\infty\) with \(\log z/\log x\ll 1\), where \(\rho_k\) is defined by a difference-differential equation and reduces to the well-known Dickman function when \(k=1\). The author establishes a \(K\)-term asymptotic expansion for \(\rho_k(u)\) with relative error \(O\{(u+k)^{-K} \}\), where \(K\) is arbitrary and \(k\geq\varepsilon(>0)\). His result is to be used in a forthcoming paper which will enlarge the range of the formula of de Bruijn and van Lint beyond that meantime established by J.-M. De Koninck and D. Hensley [J. Indian Math. Soc., New. Ser. 42(1979), 353-365 (1978; Zbl 0473.10028)]. He reports that a forthcoming paper of A. Hildebrand will study the function \(\rho_k\) for complex \(k\) with \(\text{Re}(k)>-1\).

Reviewer: G.Greaves (Cardiff)

MSC:

11N25	Distribution of integers with specified multiplicative constraints
11N37	Asymptotic results on arithmetic functions

Keywords:

representation as product of factors; difference-differential; Dickman function; asymptotic expansion

Citations:

Zbl 0131.28703; Zbl 0473.10028