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

Behrend, Felix

J. Lond. Math. Soc. 10, 42-44 (1935).

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.

Reviewer: H. M. Davenport (Cambridge)

MSC:

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

Keywords:

sequences; multiplicative number theory

Citations:

JFM 54.0090.06; Zbl 0012.05202