High-dimensional Lehmer problem on Beatty sequences

Let q be a positive integer. For each integer a with 1 < a < q and (a, q) = 1, it is clear that there exists one and only one a over bar with 1 < a over bar < q such that aa over bar = 1(q). Let k be any fixed integer with k >= 2, 0 < delta i <= 1, i = 1, 2, center dot center dot center dot , k. rn (delta 1, delta 2, center dot center dot center dot , delta k, alpha, beta, c; q) denotes the number of all k-tuples with positive integer coordinates (x1, x2, ... , xk) such that 1 <= xi <= delta iq, (xi, q) = 1, x1x2 center dot center dot center dot xk = c(q), and x1, x2, center dot center dot center dot , xk-1 E B alpha,beta. In this paper, we consider the high-dimensional Lehmer problem related to Beatty sequences over incomplete intervals and give an asymptotic formula by the properties of Beatty sequences and the estimates for hyper Kloosterman sums.

Automated summary

This paper discusses the high-dimensional Lehmer problem related to Beatty sequences over incomplete intervals and provides an asymptotic formula based on the properties of Beatty sequences and the estimates for hyper Kloosterman sums.