Arithmetical functions of the form \(f([g(n)])\) (Q1848172)

The authors remark that \(\frac 1x \sum_{n\leq x}\Omega ([n^c])=\frac 1x \sum_{n\leq x}\Omega(n)+O(1)\) as \(x\to \infty\) (where \(\Omega\) is the total number of prime factors-function) for \(c>1\) not an integer. Then they generalize this result for a composite function \(f\circ [g]\) (where [\ ] is the integer-part), by giving conditions on \(f\) and \(g\) such that \(\frac 1N\sum_{N<n\leq 2N}f([g(n)])\) behaves like \(\frac 1N \sum_{N< n\leq 2N}f(n)\) \((N\to \infty)\). Further, they obtain conditions on the positive numbers \(a\), \(b\) such that \(f([an])\) or \(f(([an],[bn]))\) are almost periodic functions. Their mean values and spectra in this case are computed. Earlier results by \textit{G. L. Watson} [Can. J. Math. 5, 451-455 (1953; Zbl 0051.03205)], as well as \textit{J. Spilker} [Arch. Math. 74, 26-29 (2000; Zbl 0887.11008)] are reobtained, as particular cases.