On the sums \(\sum _{n\leq x}A(f(n))\) and \(\sum _{p\leq x}A(f(p))\)
For complex numbers s and arithmetical functions \(\alpha\) let \(A(n)=\sum_{d| n}\alpha (d) (n/d)^ s.\) The authors prove asymptotic formulae for the sums \[ \sum_{n\leq x;\quad f(n)\neq 0}A(| f(n)|),\quad \sum_{p\leq x;\quad f(p)\neq 0}A(| f(p)|), \] where Re s\(>0\), f is a nonconstant polynomial with integer coefficients, and \(\alpha\) satisfies suitable growth conditions. Several applications are given which incidentally refine results due to \textit{R. Bellman} [Duke Math. J. 17, 159-168 (1950; Zbl 0037.312)], \textit{W. Schwarz} [Monatsh. Math. 66, 43-54 (1962; Zbl 0101.037)], and \textit{W. A. Webb} [J. Lond. Math. Soc., II. Ser. 5, 279-284 (1972; Zbl 0239.10026)].
scientific article; zbMATH DE number 3948397
scientific article