Victor Zhenyu Guo, Yuan Yi: The Lehmer problem and Beatty sequences, 471-482

Abstract:

Let $a$ and $q$ be positive integers. The D. H. Lehmer problem introduces the distribution of the set

$$\{a: a \le q, (a,q)=1, ab \equiv 1 \bmod q, 2 \nmid a+ b \}.$$

Zhang gave the initial approach. Lu and Yi considered a generalization of the Lehmer problem, which restricts the integers in short intervals. In this paper, we study a more general problem. Let

$$\ensuremath{\mathcal{B}}_{\alpha, \beta} {\mathrel{\vcenter{\base... ...\scriptsize .}}}}= \( \left\lfloor\alpha n+\beta\right\rfloor \) _{n=1}^\infty$$

be the Beatty sequence. Let $c$ be a positive integer with $(n,q)=(c, q)=1$, $0 < \delta_1, \delta_2 \le 1$. We investigate the distribution of the set

$$\{a: a \le \delta_1 q, b \le \delta_2 q, ab \equiv c \bmod q, n \nmid a+b, a \in \ensuremath{\mathcal{B}}_{\alpha, \beta} \}.$$

Key Words: The Lehmer problem, Beatty sequence, exponential sum, asymptotic formula.

2020 Mathematics Subject Classification: Primary 11B83; Secondary 11L05, 11N69.

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