On the divisor function over Piatetski-Shapiro sequences. (English) Zbl 07729527

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 [Lectures on a method in the theory of exponential sums. Berlin: Springer-Verlag (1987; Zbl 0671.10031)], 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.


11B83 Special sequences and polynomials
11L07 Estimates on exponential sums
11N25 Distribution of integers with specified multiplicative constraints
11N37 Asymptotic results on arithmetic functions


Zbl 0671.10031
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