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**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 **lim**_{n ––> oo} n^{-1} **sum**_{k = 1}^{n} 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} **sum**_{k = 0}^{l-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} **sum**_{k = 0}^{l-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 c_{1}, there is a c_{2} (c_{1}), so that if a_{1} < a_{2} < ··· < a_{k} \leq n are integers, k > c_{1}n, A = a_{1}a_{2}...a_{n}, then **sum**_{d | A} d^{-1} > c_{2} log n. The proof uses Brun's method.

Also the following result (not stated as a formal theorem) is proved: Let a_{1} < a_{2} < ··· < a_{k} \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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