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) = intx2\rho(log x/ log t)t-2 dt. Then the authors prove

sumn \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) = sump|n1 and \Omega (n) = sump\alpha||n. Then the authors prove a rather surprising result

sumn \leq x (P(n))-w(n) = \exp {(4+o(1))(log x) ½ (log log x)-1}

whereas (in contrast)

sumn \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) = sump|np and \beta(n) = sump\alpha||n\alpha p. If f(n) denotes either of these functions then

sumn \leq x1/f(n) = {1+0(\exp(-C(log x log log x) ½))} sum2 \leq n \leq x1/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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