On sequences of numbers not divisible one by another. (English) Zbl 0012.05203

The author proves that if \(a_1, a_2, \dots\) is a sequence of positive integers, no one of which divides any other, then
\[ \sum_{m\leq x} \frac1{a_i}= O\left(\frac{\log y}{\sqrt{\log\log x}}\right), \tag{1} \]
showing that the lower density of the sequence \(a_i\) is zero.
If \(m = q_1a_{i_1} = q_2a_{i_2} = \dots = q_{A(m)} a_{i_{A(m)}}\) are all the representations of \(m\) as \(qa_i\) where \(q\) is quadratfrei, then no one \(q\) divides any other \(q\) and so by a theorem of E. Sperner [Math. Z. 27, 544–548 (1928; JFM 54.0090.06)]
\[ A(m)\leq\binom{\nu(m)}{[\tfrac12\nu(m)]} \]
where \(\nu(m)\) is the number of different prime factors of \(m\). (1) is deduced by summing \(A(m)\) from 1 to \(x\).
A result of the same type as (1) has been proved by P. Erdős (see the preceding review Zbl 0012.05202), but neither result implies the other.


11B83 Special sequences and polynomials
11N25 Distribution of integers with specified multiplicative constraints
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