Multiplicative functions of special type on Piatetski-Shapiro sequences

Abstract Let f k ( n ) := ∑ d k ∣ n Φ ( d ) , $$ f_k(n):=\sum\limits_{d^k \mid n} \Phi(d), $$ where Φ( d ) is a multiplicative function, Φ( d ) = O ( d ε ). We study asymptotic behaviour of the sum T f k c ( N ) := ∑ n ≤ N f k n c , 1 < c < 2 , $$ T_{f_k}^c(N):=\sum\limits_{n \leq N} f_k\left(\left\lfloor n^c\right\rfloor\right), \quad 1<c<2, $$ and establish some asymptotic formulas for T f k c ( N ) $ T_{f_k}^c(N) $ under assumptions on f k .