On a question of Luca and Schinzel over Segal-Piatetski-Shapiro sequences (Q6171148)

Let \(\varphi\) denote the Euler totient function. Furthermore, let \(c>1\) be real, and let \((\lfloor m^c \rfloor)_{m \geq 1}\) be the Segal-Piatetski-Shapiro sequence. The authors prove that the sequence \[ \left( \sum_{m \leq n} \frac{\varphi(\lfloor m^c \rfloor)}{\lfloor m^c \rfloor} \right)_{n \geq 1} \] is dense mod 1. This extends earlier work of two of the authors [\textit{J.-M. Deshouillers} and \textit{M. Nasiri-Zare}, Springer Proc. Math. Stat. 251, 153--161 (2018; Zbl 1475.11137)] for the case of polynomials evaluated at integers. A key ingredient in the proof is the fact (established by the authors) that a Segal-Piatetski-Shapiro sequence allows local approximations by linear polynomials. As the authors point out, proving equidistribution (rather than denseness) mod 1 of the sequence under consideration remains a challenging open problem.