Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  015.24603
Autor:  Erdös, Paul
Title:  On a problem of Chowla and some related problems. (In English)
Source:  Proc. Camb. Philos. Soc. 32, 530-540 (1936).
Review:  The problem in question is whether the integers n for which d(n+1) > d(n) have density 1/2 , where d(n) denotes the number of divisors of n. The author proves that this is the case. He first proves a general theorem to the effect that if f(n) satisfies (1) f(n) \geq 0, (2) f(mn) = f(m)+f(n) provided that (m,n) = 1, (3) sum f(p)/p (summed over all primes p) converges, then the density of the integers n for which f(n+1) < f(n) is 1/2 . The method of proof is based on that used by the author in a previous paper (see Zbl 012.01004). The fundamental idea is that of approximating to f(n) by

fk(n) = sump < pk f(p\alpha), where p\alpha | m, p\alpha+1 \nmid m.

The author then establishes that the result of the theorem holds also for V(n), the number of different prime factors of n, which satisfies (1) and (2) but not (3). The proof of this is on the same lines but much more complicated, as k is taken to be a function of n of the order of magnitude n(log log n)^{-3}. Finally the result for d(n) is obtained from that for V(n) by proving that for almost all n,

(d(n+1)-d(n)) (V(n+1)-V(n)) > 0.

Reviewer:  Davenport
Classif.:  * 11N25 Distribution of integers with specified multiplicative constraints
Index Words:  Algebra, number theory

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