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**Zentralblatt MATH**

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

**Zbl.No: ** 419.10042

**Autor: ** Alladi, K.; Erdös, Paul

**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 p_{1}(n), p_{2}(n)... defined as the biggest prime factor of n, and the next smaller prime factor and so on. More precisely let n = **prod**_{i = 1}^{r}p_{i}^{\alphai} where p_{1} > p_{2} > ... and \alpha_{i} > 0. A(n) = \Sigma\alpha_{1}, A^*(n) = \Sigma p_{i}, Omega(n) = \Sigma\alpha_{1}, \omega(n) = r, P_{1}(n) = P_{1}^*(n) = p_{1}, P_{k}^*(n) = p_{k} for k \leq \omega(n) and zero for k > \omega(n), P_{k}(n) = P_{1}**(**\frac n{P_{1}(n)P_{2}(n)... P_{k-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), P_{k}(n) and P_{k}^*(n). With this in view they introduce S_{1}(x,k) = **sum**_{2 \leq n \leq x} \frac{A(n)-P_{1}(n)- ... -P_{k-1}(n)}{P_{1}(n)}, S_{2}(x,k) = **sum**_{2 \leq n \leq x}\frac{A^*(n)-P_{1}^*(n)-...-P_{k-1}^*(n)}{P_{1}(n)},

S_{3}(x,k) = **sum**_{2 \leq n \leq x} \frac{P_{k}(n)}{P_{1}(n)}, and S_{4}(k,x) = **sum**_{2 \leq n \leq x} \frac{P_{k}^*(n)}{P_{1}(n)}, where k \geq 1 is any fixed positive integer. In anearlier paper they proved that as x ––> oo,

**sum**_{1 \leq n \leq x}(A(n)-P_{1}(n)-...-P_{k-1}(n)) ~ **sum**_{1 \leq n \leq x}P_{k}(n) ~ **sum**_{1 \leq n \leq x}\P_{k}^*(n) ~ **sum**_{1 \leq n \leq x}(A^*(n)-P_{1}^*(n)-...-\P_{k-1}^*(n)) ~ a_{k}(log x)^{-k}x^{1+1/k}, where a_{k} is a positive constant depending on k. They also proved

**sum**_{1 \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,

S_{1}(x,k) ~ S_{2}(x,k) ~ S_{3}(x,k) ~ S_{4}(x,k) ~ a_{k}'(log x)^{1-k}x, where a_{k}' 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), P_{1}(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