On the average value for the number of divisors of numbers of form \(ab+1\)
Let \(A\), \(B\) be two sets of distinct integers in the interval \([1,x ]\). Various authors have studied the prime factor or divisibility structure of the sums \(a+b\) and, more recently, \(ab+1\), where \(a\in A\), \(b\in B\). In [Ill. J. Math. 38, No. 1, 1-18 (1994; Zbl 0793.11025)], \textit{A. Sárközy} and \textit{C. L. Stewart} obtained a lower bound for \((|A||B|)^{-1} \sum_{a\in A} \sum_{b\in B} \tau(a+ b)\). The object of this paper is to study the corresponding quantity with summand \(\tau (ab+1)\). However the undue influence of any small prime factors of \(ab+1\) renders the establishment of an asymptotic formula impossible in general, as the author illustrates by means of examples of sets \(A\), \(B\). To avoid this difficulty, he considers instead the sum \[ T_K=( |A||B|)^{-1} \sum_{a\in A} \sum_{b\in B} \tau_K (ab+1) \] where \(\tau_K (n)\) denotes the number of positive divisors \(d\) of \(n\) such that every prime factor of \(d\) exceeds \(K\). He proves the expected result \[ T_K\sim \prod_{p\leq K} (1- {\textstyle {1\over p}}) \log x \] when \(\log K\leq (\log x)^{1\over 2}\), but \(K\) is not too small in comparison with \(U= x^2/|A||B|= o(\log x)\) in a sense that is made explicit.
scientific article; zbMATH DE number 755504
scientific article