On coprime elements of the Beatty sequence

Abstract

This note discusses two applications of the asymptotic formula obtained by the authors for the number of values of the Beatty sequence in an arithmetic progression with increasing difference: asymptotic formulas are obtained for the number of elements of the Beatty sequence that are coprime to the (possibly growing) natural number 𝑎, as well as for the number of pairs of coprime elements of two Beatty sequences. Here are the main results.
Let 𝛼 > 1 be an irrational number and 𝑁 be a sufficiently large natural number. Then if the partial quotients of the continued fraction of the number 𝛼 are limited, then for the number 𝑆_𝛼,𝑎(𝑁) of elements of the Beatty sequence [𝛼𝑛], 1 ⩽ 𝑛 ⩽ 𝑁, coprime to the number 𝑎,
the following asymptotic formula holds

$$
S_{\alpha,a}(N)=N\frac{\varphi(a)}{a} + O\left(\min(\sigma(a)\ln^3 N, \sqrt{N}\tau( a)(\ln\ln N)^3)\right),
$$

where 𝜏 (𝑎) is the number of divisors of 𝑎 and 𝜎(𝑎) is the sum of the divisors of 𝑎.
Let 𝛼 > 1 and 𝛽 > 1 be irrational numbers and 𝑁 be a sufficiently large natural number. Then if the incomplete quotients of continued fractions of the numbers 𝛼 and 𝛽 are bounded, then for the number 𝑆_𝛼,𝛽(𝑁) of pairs of coprime elements of Beatty sequences [𝛼𝑚], 1 ⩽ 𝑚 ⩽ 𝑁, and [𝛽𝑛], 1 ⩽ 𝑛 ⩽ 𝑁, the following asymptotic formula holds

$$
S_{\alpha,\beta}(N)=\frac{6}{\pi^2}N^2 + O\left(N^{3/2}(\ln\ln N)^6 \right).
$$