On general divisor functions over Piatetski-Shapiro sequences

Abstract

In this paper, we consider the general divisor functions over Piatetski-Shapiro sequences. We can give some general results which contain some special divisor functions. Precisely, we extend the divisor problem over Piatetski-Shapiro sequences to the function $f(n),$ where $f(n)\ll n^{\varepsilon},$ $$f(n)=\sum_{n=n_{1}n_{2}} \tau(n_{1})g(n_{2}),$$ $\tau(n)$ is the number of representations of $n$ as product of two natural numbers and \[ \sum_{1\leq n\leq x}|g(n)|\ll x^{5/8+\varepsilon}. \] On the other hand, we also considered these arithmetic functions over Piatetski-Shapiro sequences in arithmetic progressions.