On the estimation of the second central moment for any additive arithmetic functions
Let \(f(m)\) be an additive arithmetic function, \[ A_ n(f)=\sum_{p\leq n}f(p)p^{-1},\quad D^ 2_ n(f)=\sum_{p^{\alpha}\leq n}| f^ 2(p^{\alpha})| \, p^{-\alpha}, \] \[ \tau_ n(f)=(D^ 2_ n(f)/n)\sum^{n}_{m=1}| f(m)-A_ n(f)|^ 2. \] \(\tau_ n\) be the supremum over all \(f\neq 0\) then \[ \tau_ n=1.5+O(\log^{-1/2}n). \] The same estimate is also obtained in the cases \(A_ n(f)=\sum_{p^{\alpha}\leq n}f(p^{\alpha})(1-p^{-1})p^{-\alpha}\) and \(A_ n(f)=\sum_{p^{\alpha}\leq n}f(p^{\alpha})p^{-\alpha}.\) Only estimates from above and below for \(\tau_ n\) were known earlier.
scientific article; zbMATH DE number 3841993