Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 010.10303
Autor: Erdös, Paul
Title: On the density of the abundant numbers. (In English)
Source: J. London Math. Soc. 9, 278-282 (1934).
Review: This paper gives a new and elementary proof that A(n)/n tends to a limit as n ––> oo where A(n) is the number of abundant numbers \leq n (see H.Davenport Zbl 008.19701; S.Chowla Zbl 010.00803). It is proved that the series of reciprocals of the primitive abundant numbers converges, and from this the result follows immediately. It is first shown that there are only o(n/(log n)2) primitve abundant numbers m \leq n which do not (a) have a simple prime divisor pm between (log n)10 and n1 \over 40 log log n, (b) satisfy 2 \leq {\sigma(m) \over m} \leq 2+{2 \over {n1\over 20 log log n}}, (1) [\sigma(m) = sum of divisors of m]. Further, the number of integers m \leq n which do have properties (a) and (b) is O(n/(log n)10). For let m = pmlm, then lm < n/(log n)10, and it suffices to prove that lm\ne lm', when m\ne m'. Now \lm = lm', pm < pm' would imply
{\sigma (m)\over m} {m'\over \sigma(m')} = {(pm+1) \over pm} {pm'\over (pm'+1)} \geq 1+{1 \over pm(pm'+1)} > 1+{1 \over {n1\over 20 log log n}}, which would contradict (1).
Reviewer: Davenport (Cambridge)
Classif.: * 11A25 Arithmetic functions, etc.
11N25 Distribution of integers with specified multiplicative constraints
Index Words: algebra, number theory
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