Topical problems concerning~Beatty~sequences

A. V. Begunts ; D. V. Goryashin

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 97-106.

Résumé

In English-language literature, Beatty sequence means a sequence of the form $[\alpha n]$ and, more generally, $[\alpha n+\beta]$, where $\alpha$ is a positive irrational number, $\beta$ is a real number (if $\beta=0$, then the sequence is called homogeneous, otherwise it is called non-homogeneous). In Russian literature, such sequences are usually referred to as greatest-integer sequences of a special form, or as generalized arithmetic progressions. The properties of these sequences have been under extensive study ever since late 19th century and up to nowadays. This paper contains a review of main directions in Beatty sequences research, and points out some key results. The investigation of the distribution of prime numbers in Beatty sequences, once started in 1970s, was continued in 2000s, when due to application of new methods it became possible to improve estimates of remainder terms in asymptotic formulas. A wide range of tasks deal with sums of the values of arithmetical functions over Beatty sequences. Various authors obtained asymptotic formulas for sums of the values of divisor function $\tau(n)$ and multidimensional divisor function $\tau_k(n)$, of divisor-summing function $\sigma(n)$, of Euler function $\varphi(n)$, of Dirichlet characters, of prime divisor counting function $\omega(n)$. Besides that, there appeared various results concerning quadratic residues and nonresidues in Beatty sequences. Since 1990s additive tasks associated with Beatty sequences became a topical direction of study. Some analogues of classical Goldbach-type problems, where primes belong to Beatty sequences, are under research, along with tasks of representation of integers as a sum, a part of summands of which are members of such a sequence.

Détail

Keywords: Beatty sequences, integer sequence, prime numbers, mean value of a number-theoretic function, sums.

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