Received August 20, 2020. Published online October 14, 2021.
Abstract: 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.
Keywords: arithmetic progression; character sum; exponent pair method; square-full number
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Affiliations: Angkana Sripayap, Department of Mathematics, Faculty of Science, Kasetsart University, 50 Thanon Ngam Wong Wan, Lat Yao, Chatuchak, Bangkok 10900, Thailand, e-mail: fscianr@ku.ac.th; Pattira Ruengsinsub (corresponding author), Department of Mathematics, Faculty of Science, Kasetsart University, 50 Thanon Ngam Wong Wan, Lat Yao, Chatuchak, Bangkok 10900, Thailand, e-mail: fscipar@ku.ac.th; Teerapat Srichan, Department of Mathematics, Faculty of Science, Kasetsart University, 50 Thanon Ngam Wong Wan, Lat Yao, Chatuchak, Bangkok 10900, Thailand, e-mail: fscitrp@ku.ac.th
