On coprime elements of the Beatty sequence

A. V. Begunts ; D. V. Goryashin

Čebyševskij sbornik, Tome 25 (2024) no. 1, pp. 164-169.

Résumé

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 $\alpha>1$ be an irrational number and $N$ be a sufficiently large natural number. Then if the partial quotients of the continued fraction of the number $\alpha$ are limited, then for the number $S_{\alpha,a}(N)$ of elements of the Beatty sequence $[\alpha n]$, $1\leqslant n\leqslant N$, 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 $\tau(a)$ is the number of divisors of $a$ and $\sigma(a)$ is the sum of the divisors of $a$. Let $\alpha>1$ and $\beta>1$ be irrational numbers and $N$ be a sufficiently large natural number. Then if the incomplete quotients of continued fractions of the numbers $\alpha$ and $\beta$ are bounded, then for the number $S_{\alpha,\beta}(N)$ of pairs of coprime elements of Beatty sequences $[\alpha m]$, $1\leqslant m\leqslant N$, and $[\beta n]$, $1\leqslant n\leqslant N$, the following asymptotic formula holds $$ S_{\alpha,\beta}(N)=\frac{6}{\pi^2}N^2 + O\left(N^{3/2}(\ln\ln N)^6 \right). $$

Détail

Keywords: Beatty sequence, coprime numbers, asymptotic formula.

Bibliographie

[1] Begunts, A. V., Goryashin, D. V., “On the values of Beatty sequence in an arithmetic progression”, Chebyshevskii Sbornik, 21:1 (2020), 364–367 | DOI | DOI | MR

[2] Begunts, A. V., Goryashin, D. V., “Topical problems concerning Beatty sequences”, Chebyshevskii Sbornik, 18:4 (2017), 97–105 | DOI | MR

[3] Vinogradov, I. M., Fundamentals of Number Theory, Nauka, M., 1981, 176 pp. | MR