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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.

MSC:

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

Citations:

Zbl 0671.10031
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