## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  419.10042
Title:  On the asymptotic behavior of large prime factors of integers. (In English)
Source:  Pac. J. Math. 82, 295-315 (1979).
Review:  The paper under review continues the work of Hardy and Ramanujan on round numbers and related topics. The authors study the functions p1(n), p2(n)... defined as the biggest prime factor of n, and the next smaller prime factor and so on. More precisely let n = prodi = 1rpi\alphai where p1 > p2 > ... and \alphai > 0. A(n) = \Sigma\alpha1, A^*(n) = \Sigma pi, Omega(n) = \Sigma\alpha1, \omega(n) = r, P1(n) = P1^*(n) = p1, Pk^*(n) = pk for k \leq \omega(n) and zero for k > \omega(n), Pk(n) = P1(\frac n{P1(n)P2(n)... Pk-1(n)}) for 1 < k \leq \Omega(n) and zero for k > \Omega(n). The functions \Omega(n) and \omega(n) were studied by Hardy and Ramanujan who proved that their normal order is log log n. The authors make a comparative study of A(n), A^*(n), Pk(n) and Pk^*(n). With this in view they introduce

S1(x,k) = sum2 \leq n \leq x \frac{A(n)-P1(n)- ... -Pk-1(n)}{P1(n)},   S2(x,k) = sum2 \leq n \leq x\frac{A^*(n)-P1^*(n)-...-Pk-1^*(n)}{P1(n)},

S3(x,k) = sum2 \leq n \leq x \frac{Pk(n)}{P1(n)}, and S4(k,x) = sum2 \leq n \leq x \frac{Pk^*(n)}{P1(n)},

where k \geq 1 is any fixed positive integer. In anearlier paper they proved that as x ––> oo,

sum1 \leq n \leq x(A(n)-P1(n)-...-Pk-1(n)) ~ sum1 \leq n \leq xPk(n) ~ sum1 \leq n \leq x\Pk^*(n) ~ sum1 \leq n \leq x(A^*(n)-P1^*(n)-...-\Pk-1^*(n)) ~ ak(log x)-kx1+1/k,

where ak is a positive constant depending on k. They also proved

sum1 \leq n \leq x(A(n)-A^*(n)) = x log log x+0(x).

To these they add another set of interesting results. Namely that as x ––> oo,

S1(x,k) ~ S2(x,k) ~ S3(x,k) ~ S4(x,k) ~ ak'(log x)1-kx,

where ak' is a positive constant depending only on k. These results give satisfactory information of the asymptotic behaviour (i.e. average order, normal order etc.) of the functions which they consider. For instance it is somewhat surprising that A(n), A^*(n), P1(n) are almost always nearly of the same order. But none of them possess a normal order. The last mentioned result is deduced from the results mentioned above by appealing to a result of P.D.T.A.Elliott.
Reviewer:  K.Ramachandra
Classif.:  * 11N37 Asymptotic results on arithmetic functions
11N05 Distribution of primes
Keywords:  Hardy-Ramanujan theorem; round numbers; large prime factors; average order; normal order

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag