Geodesic
On the divisor function over Piatetski-Shapiro sequences
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 613-620

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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 $1$, $$ \sum _{n\leq x} d([n^c])= cx\log x +(2\gamma -c)x+O\Bigl (\frac {x}{\log x}\Bigr ), $$ 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.
DOI : 10.21136/CMJ.2023.0205-22
Classification : 11B83, 11L07, 11N25, 11N37
Keywords: divisor function; Piatetski-Shapiro sequence; exponential sum 
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Wang, Hui; Zhang, Yu. On the divisor function over Piatetski-Shapiro sequences. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 613-620. doi: 10.21136/CMJ.2023.0205-22

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