Some remarks on the [x/n]-sequence — Yuta Suzuki (Rikkyo University)

Bordellès--Dai--Heyman--Pan--Shparlinski introduced the study of mean values of arithmetic functions over the sequence [x/n], where n runs over positive integers up to x. Despite its funny appearance as if one written down half asleep, several authors have become interested in such sums and indeed, they can be connected to exponential sums of rather conventional form and some of such sums are rather difficult to deal with. The numbers [x/n] without counting the multiplicity of n have been also studied by several authors.

In this talk, we give some remarks on the [x/n]-sequence:
1. An improvement of a result of Wu--Yu on the distribution of the
[x/n] sequence in arithmetic progressions.
2. On the distribution of coprime pairs of the [x/n] numbers.
3. A "multiplicative analog" of the Titchmarsh divisor problem.

We also remark that the idea of the second topic above gives:
2-1. A solution of an open problem of Banks--Shparlinski on the coprime pair of palindromes of fixed length.
2-2. Distribution of coprime pairs from the Piatetski-Shapiro sequence of any order,
which improves the result of Srichan restricted to the Piatetski-Shapiro sequence of order < 3/2.
(This talk is based on a joint work with Kota Saito, Wataru Takeda, Yuuya Yoshida and a joint work with Hirotaka Kobayashi and Ryota Umezawa.)