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Keywords:
divisor function; Piatetski-Shapiro sequence; exponential sum
divisor function; Piatetski-Shapiro sequence; exponential sum
Summary:
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 $1<c<\frac 65$, $$ \sum _{n\leq x} d([n^c])= cx\log x +(2\gamma -c)x+O\Bigl (\frac {x}{\log x}\Bigr ), $$ 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.
References:
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