On an exponential D. H. Lehmer problem

Abstract

For an odd prime $ p $ and a positive integer $ \alpha $, let $ g $ be of multiplicative order $ \tau $ modulo $ q $ and $ q = p^{\alpha} $. Denote by $ N(h, g, q) $ the number of $ a $ such that $ h\nmid (a+(g^{a})_{q}) $ for any $ 1\leq a\leq \tau $ and a fixed integer $ h\geq 2 $ with $ (h, q) = 1 $. The main purpose of this paper is to give a sharp asymptotic formula for $ N(k, h, g, q) = \mathop{\sum\limits_{\begin{smallmatrix} a = 1\\ h\nmid (a+(g^{a})_{q}) \end{smallmatrix}}^{\tau}}\left|a-(g^{a})_{q}\right| ^{2k} $ where $ k $ is any nonnegative integer and $ (a)_{q} $ denotes the smallest positive residue of $ a $ modulo $ q $. In addition, we know that $ N(h, g, q) = N(0, h, g, q) $.

Keywords

• primitive roots
• exponential lehmer problem
• exponential functions
• asymptotic formula