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arXiv · 2104.11701

On a question of Luca and Schinzel over Segal-Piatetski-Shapiro sequences

Jean-Marc Deshouillers·Mehdi Hassani·Mohammad Nasiri-Zare
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Abstract

We extend to Segal-Piatetski-Shapiro sequences previous results on the Luca-Schinzel question over integral valued polynomial sequences. Namely, we prove that for any real $c$ larger than $1$ the sequence $(\sum_{m\le n} \varphi(\lfloor m^c \rfloor) /\lfloor m^c \rfloor)_n$ is dense modulo $1$, where $\varphi$ denotes Euler's totient function. The main part of the proof consists in showing that when $R$ is a large integer, the sequence of the residues of $\lfloor m^c \rfloor$ modulo $R$ contains blocks of consecutive values which are in an arithmetic progression.

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Jean-Marc Deshouillers, Mehdi Hassani, Mohammad Nasiri-Zare. 2021-04-23. On a question of Luca and Schinzel over Segal-Piatetski-Shapiro sequences. https://arxiv.org/abs/2104.11701

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