Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  147.30201
Autor:  Erdös, Pál
Title:  Some remarks on number theory. II (In English)
Source:  Isr. J. Math. 5, 57-64 (1967).
Review:  [Part I cf. Zbl 131.03902]
According to the author's summary, this paper contains several disconnected remarks on number theory. The main results are:
Theorem 1. Let f(k) be a real-valued arithmetical function, with

limn ––> oo n-1 sumk = 1n f(k) = \alpha(\ne ±oo).

Assume that for every \eta > 0, there is a g(\eta) so that for l > g(\eta) and n > 0, l-1 sumk = 0l-1 f(n+k) < \alpha+\eta; then to every \epsilon > 0, \delta > 0 , there is an h(\epsilon,\delta) so that for all but \epsilon x integers n < x, we have for every l > h(\epsilon,\delta) that

|l-1 sumk = 0l-1 f(n+k)-\alpha | < \delta.

This generalizes (a strengthened form of) a result of R.Bellman and H.N.Shapiro (Zbl 057.28602).
Theorem 2. To every c1, there is a c2 (c1), so that if a1 < a2 < ··· < ak \leq n are integers, k > c1n, A = a1a2...an, then sumd | A d-1 > c2 log n. The proof uses Brun's method.
Also the following result (not stated as a formal theorem) is proved: Let a1 < a2 < ··· < ak \leq x be k integers such that no two of them are relatively prime, but every three are. If, for given x, one sets max k = f(x), then f(x) = (1/2 +o(1))(log x)/(log log x).
Reviewer:  E.Grosswald
Classif.:  * 11N64 Characterization of arithmetic functions
Index Words:  number theory

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