The value distribution of a multiplicative function (Q1605544)

The author considers a family of multiplicative functions \(\kappa_{\alpha}(n)\) defined by \(\kappa_\alpha(p^k)= (\alpha^{k+1}- (-1)^{k+1})/ (k+1)\), where \(0< \alpha\leq 2\), \(\alpha\neq 1\) and gives an asymptotic formula for the sum \(\sum_{n\leq x}\kappa_{\alpha}(n)\) as \(x\to\infty\): \[ \sum_{n\leq x}\kappa_{\alpha}(n)= A_0(\alpha)\zeta(2)\frac{1}{\Gamma(\alpha-1)} \frac{x}{(\log x)^{2-\alpha}}+O\left(\frac{x}{(\log x)^{3-\alpha}}\right),\;0<\alpha<2,\;\alpha\not=1; \] \[ \sum_{n\leq x}\kappa_2(n)= A\zeta(2)x\log x+O(x), \] where \(A_0(\alpha)\) is a known function and \(A\) is a constant. The proof uses contour integration and an estimate for the zero-free region of \(\zeta(s)\).