Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  010.39103
Autor:  Erdös, Paul
Title:  On primitive abundant numbers. (In English)
Source:  J. London Math. Soc. 10, 49-58 (1935).
Review:  The author has proved in a previous paper (see Zbl 010.10303) that the number N (n) of primitive abundant numbers \leq n is O (n/ log2 n). He proves in this paper the striking result that for large n,

ne-c1,x < N(n) < ne-c2x

where x = \sqrt{log n log log n}, and c1, c2 are absolute constants (say 8, {1 \over 2 5}
Define for any m

sm = prod \Sb{p|m}
{p2\nmid m}\endSb qm = {m\over sm}.

The author proves that all but O(ne-c3x) of the primitive abundant numbers m \leq n satisfy: (1) qm < e 1/8 x, (2) the greates prime factor of m is > ex. It is then shown that these primitive abundant numbers satisfy also (a) sm has a divisor Dm between 1/2 e 1/2 x and 1/2 e 1/8 x, (b) 2 \leq \sigma (m)/m < 2+2e^{-x}. [\sigma (m) = sum of divisors of m]. Now, as in the previous paper, it follows that the numbers sm/Dm are all different and \leq 2n e- 1/8 x. This gives the upper bound for N(n).
To obtain the lower bound, the author considers numbers of the form 2lp1...pk, where p1,...pk are any k different primes between (k-1)2l+1 and k2l+1, and

ex-4 < 2l < ex-3,   k = [\sqrt{log n \over log log n} ]-2.

These numbers are all primitive abundant, and an application of the prime-number theorem shews that there areat least ne-8x of them.
Reviewer:  Davenport (Cambridge)
Classif.:  * 11A25 Arithmetic functions, etc.
                   11N25 Distribution of integers with specified multiplicative constraints
Index Words:  number theory

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