On generalized square-full numbers in an arithmetic progression

summary:Let $a$ and $b\in \mathbb {N}$. Denote by $R_{a,b}$ the set of all integers $n>1$ whose canonical prime representation $n=p_1^{\alpha _1}p_2^{\alpha _2}\cdots p_r^{\alpha _r}$ has all exponents $\alpha _i$ $(1\leq i\leq r)$ being a multiple of $a$ or belonging to the arithmetic progression $at+b$, $t\in \mathbb {N}_0:=\mathbb {N}\cup \{0\}$. All integers in $R_{a,b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given