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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 776.11013

**Autor: ** Erdös, Paul; Zhang, Zhenxiang

**Title: ** Upper bound of **sum** 1/(a_{i} log a_{i}) for primitive sequences. (In English)

**Source: ** Proc. Am. Math. Soc. 117, No.4, 891-895 (1993).

**Review: ** A sequence *A* = **{**a_{i}**}** of positive integers a_{1} < a_{2} < ··· is called primitive if a_{i}\nmid a_{j} for i\ne j. Define f(*A*) = **sum**_{a in A} (1/a log a). In 1935, the first author proved that there exists an absolute constant c such that f(*A*) < c for any primitive sequence *A*. The main result of this paper is that c = 1.84 is admissible. The authors also give a necessary and sufficient condition for a more recent conjecture of the first author namely that for any primitive sequence *A*, **sum**_{a \leq n,a in A}{1\over a log a} \leq **sum**_{p \leq n}{1\over p log p} (n > 1), where p denotes a prime number.

**Reviewer: ** M.Nair (Glasgow)

**Classif.: ** * 11B83 Special sequences of integers and polynomials

**Keywords: ** upper bound; primitive sequence

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