Czechoslovak Mathematical Journal, Vol. 73, No. 2, pp. 613-620, 2023
On the divisor function over Piatetski-Shapiro sequences
Hui Wang, Yu Zhang
Received May 15, 2022. Published online March 6, 2023.
Abstract: Let $[x]$ be an integer part of $x$ and $d(n)$ be the number of positive divisor of $n$. Inspired by some results of M. Jutila (1987), we prove that for $1<c<\frac65$, $\sum_{n\leq x} d([n^c])= cx\log x +(2\gamma-c)x+O (\frac{x}{\log x}),$ where $\gamma$ is the Euler constant and $[n^c]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.
Keywords: divisor function; Piatetski-Shapiro sequence; exponential sum
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Affiliations: Hui Wang (corresponding author), Yu Zhang, School of Mathematics of Shandong University, 27 Shanda Nanlu, Jinan 250100, Shandong, P. R. China, e-mail: wh0315@mail.sdu.edu.cn, yuzhang0615@mail.sdu.edu.cn
