The distribution of the average prime divisor of an integer - Springer Link
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The distribution of the average prime divisor of an integer - Springer Link
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The distribution of the average prime divisor of an integer By JEAN-MARIEDE KONINCKand ALEKSANDARIVI~
1. Introduction. Let us usual ~o(n) and (~(n) denote the number of distinct prime divisors and the total number of prime divisors of n respectively, and let P (n) denote the largest prime factor of n => 2. One may define the average prime factor of n as (1.1)
P,(n)-
/~(n) co(n)'
fl(n) = 52p,
pl.
B(n)
P*(n) - szt )'-"n" B(n) = 52 ~p, p~ll, where as usual p~ IIn means that p~ divides n (p prime), but p~+ 1 does not. Thus one may consider P, (n) as the average of distinct prime factors of n, while P* (n) is the average of all prime factors of n, counted with their respective multiplicities. The functions fl(n) and B (n) are additive, and they are "large" in the sense of Chapter 6 of [4]. Problems involving fl(n) and B(n) have attracted much attention in recent years. Thus in [1] K. Alladi and P. Erdrs showed (1.2)
g2X2
(1.3)
•
P(n)~
2 1, when one used the asymptotic formula (4.3) of [4]. This proves Theorem 3. 4. Remarks. Instead of summing P, (n) or P* (n) one may sum their reciprocals, and this was investigated in [7] and [8]. In that case one has, following the proof of (1.7) in [8], 1 co(n) '1" f log x ~1/2 1 2 - 2=
