On sums involving the Euler totient function (Q6564341)

For any fixed integer \(k\geq 2\) and any \(x>3\) let\N\[\NS_k(x):=\sum _{n_{1}\cdots n_{k}\leq x}\varphi (\gcd (n_{1},\ldots ,n_{k})),\N\]\Nwhere \(\varphi\) is the Euler function and \(\gcd (n_{1},\ldots ,n_{k})\) denote the greatest common divisor of positive integers \(n_{1},\ldots ,n_{k}\). In the paper under review, the authors prove that\N\[\NS_2(x)=c_1x\log^2 x+c_2x\log x+c_3x+O(x^{55/84+\varepsilon}),\N\]\Nwhere \(c_1=1/(4\zeta^2(2))\) and the constants \(c_2\) and \(c_3\) computed explicitly in terms of \(\zeta(2)\), \(\zeta'(2)\), \(\zeta''(2)\), the Euler constant and the first Stieltjes constant. They also give precise approximations for \(S_{3}(x)\), \(S_{4}(x)\), \(S_{5}(x)\) and provide a conditional approximation for \(S_k(x)\) (\(k\geq 5\)) of the form\N\[\NS_k(x)=xP_{\varphi,k}(\log x)+O(x^{(k-1)/2k+\varepsilon}),\N\]\Nwhere \(P_{\varphi,k}\) is a polynomial of degree \(k-1\) depending on \(\varphi\).