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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 615.10055

**Autor: ** Erdös, Paul; Ivic, A.; Pomerance, C.

**Title: ** On sums involving reciprocals of the largest prime factor of an integer. (In English)

**Source: ** Glas. Mat., III. Ser. 21(41), 283-300 (1986).

**Review: ** Let P(n) denote the largest prime factor of n. Let \rho(u) denote the continuous solution to the differential delay equation u\rho'(u) = -\rho(u-1) with the initial condition \rho(u) = 1 for 0 \leq u \leq 1. (\rho(u) is usually called the Dickman-de Bruijn function). Let \delta(x) = **int**^{x}_{2}\rho(log x/ log t)t^{-2} dt. Then the authors prove **sum**_{n \leq x}(P(n))^{-1} = x\delta (x)(1+O((log log x/ log x)^{ ½})). The authors prove many other interesting results. We quote one or two results. Let w(n) = **sum**_{p|n}1 and \Omega (n) = **sum**_{p\alpha}||n. Then the authors prove a rather surprising result

**sum**_{n \leq x} (P(n))^{-w(n)} = \exp **{**(4+o(1))(log x)^{ ½} (log log x)^{-1}**}** whereas (in contrast)

**sum**_{n \leq x} (P(n))^{-\Omega (n)} = log log x+D+O(1/ log x) where D is a constant.

Another result which I would like to quote is about K. Alladi's functions \beta(n) = **sum**_{p|n}p and \beta(n) = **sum**_{p\alpha}||n\alpha p. If f(n) denotes either of these functions then

**sum**_{n \leq x}1/f(n) = **{**1+0(\exp(-C(log x log log x)^{ ½}))**}** **sum**_{2 \leq n \leq x}1/P(n). The reader would find many other interesting results.

**Reviewer: ** K.Ramachandra

**Classif.: ** * 11N05 Distribution of primes

11N37 Asymptotic results on arithmetic functions

**Keywords: ** reciprocals of largest prime factor; Dickman-de Bruijn function

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