On the average value for the number of divisors of sums \(a+b\) (Q1320957)

The authors prove three theorems on the average value of \(\tau(a+b)\) as \(a\) and \(b\) run over sufficiently dense subsets \({\mathcal A}\) and \({\mathcal B}\) of \(\mathbb{N}\). Theorem 1: Let \(\varepsilon>0\), \(N\in\mathbb{N}\) and \({\mathcal A},{\mathcal B} \subseteq \{1,2,\dots,N\}\) with \(\min (|{\mathcal A}|, |{\mathcal B}|)> \varepsilon N\). There exist effectively computable positive constants \(c_ 0\), \(c_ 1\) and \(c_ 2\) such that if \(N>c_ 0\) and \(\exp(-c_ 1(\log N)^{1/2})\) \(< \varepsilon< 1/8\), then \[ {1\over {|{\mathcal A}|\;|{\mathcal B}|}} \sum_{\textstyle {{a\in {\mathcal A}},{b\in {\mathcal B}}}} \tau(a+b)> {{c_ 2 \log N} \over {(\log(1/\varepsilon))^ 5 \log\log (1/\varepsilon)}}. \] In the opposite direction, it is shown that the following is true. Theorem 2: There exists effectively computable positive constants \(c_ 3\), \(c_ 4\) and \(c_ 5\) such that if \(N>c_ 3\) and \(\varepsilon\) satisfies \(\exp(-\log N/\log \log N)< \varepsilon< c_ 4\), then there exists \({\mathcal A}\subseteq \{1,2,\dots, N\}\) with \(|{\mathcal A}|> \varepsilon N\) for which \[ {1\over {|{\mathcal A}|^ 2}} \sum_{a,a'\in{\mathcal A}} \tau(a+a')< {{c_ 5\log N} \over {\log\log (1/\varepsilon)}}. \] Finally, Theorem 3. For each \(\delta>0\), there exist positive \(c_ 6\) and \(c_ 7\), effectively computable in terms of \(\delta\), such that if \(N>c_ 6\) and \(\varepsilon\) satisfies \(N^{-1/8}< \varepsilon< c_ 7\), then there exists \({\mathcal A}\subseteq \{1,2,\dots, N\}\) with \(| {\mathcal A}|> \varepsilon N\) for which \[ {1\over {|{\mathcal A}|^ 2}} \sum_{a,a'\in{\mathcal A}} \tau(a+a')> \Biggl(\exp\Biggl((1- \delta)\log 2\log \biggl({1\over\varepsilon} \biggr)\Biggl/\log \log \biggl({1\over \varepsilon}\biggr) \Biggr)\Biggr)\log N. \] The authors conjecture that this last lower bound is almost optimal in the sense that the \(-\delta\) in it cannot be replaced by \(+\delta\).