On general divisor functions over Piatetski-Shapiro sequences

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 $f(n)\ll n^{\epsilon },$ $$f(n)=\sum _{n=n_1{n}_2}\tau (n_1)g(n_2),$$ $\tau (n)$ is the number of representations of $n$ as product of two natural numbers and $$\sum _{1\le n\le x}\mathrm{\mid }g(n)\mathrm{\mid }\ll x^{5\mathrm{/}8+\epsilon }\mathrm{.}$$ On the other hand, we also considered these arithmetic functions over Piatetski-Shapiro sequences in arithmetic progressions.