On product representation of powers. I
The authors study the solvability of the equation (1) \(a_1 a_2\cdots a_k= z^2\), \(a_1, a_2,\dots, a_k\in A\), \(a_1< a_2<\cdots< a_k\), \(x\in \mathbb{N}\) for fixed \(k\) and dense sets \(A\) of natural numbers. It is shown that if \(k\) is even, \(k\geq 4\), and if \(A\) is of positive density then the above equation is solvable. In particular, it is proved that if \[ \begin{aligned} F_k(n) & := \max_{A\subseteq [1,n]; (1)\text{ is not solvable for } A}| A|, \text{ and }\\ L_k(n) & := \max_{A\subseteq [1,n]; (1)\text{ is not solvable for }A} \sum_{a\in A} {1\over a}\end{aligned} \] then for all \(n\in \mathbb{N}\), \(F_2(n)\) is equal to the number of square-free integers not exceeding \(n\), that is \(F_2(n)\sim 6/\pi^2\cdot n\). Non-trivial upper and lower bounds for \(F_k(n)\) for \(k= 3,4,4j, 4j+ 2\) are established and it is shown that \((\log 2-\varepsilon)n < F_{2k+ 1} < n-(1- \varepsilon) n(\log n)^{-2}\) (the authors remark that the above lower bound could be improved slightly by a lengthy computation). Moreover, the following results on \(L_k\) are established for \(n\to \infty\): \(L_{4j} = (1+ o(1))\log\log n\), \(L_{4j+ 2} = (3/2+ o(1))\log\log n\), \(L_{2j+ 1} = 1+(1/2+ o(1))\log n\).
scientific article; zbMATH DE number 823417