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) $.
Citation: Zhefeng Xu, Jiankang Wang, Lirong Zhu. On an exponential D. H. Lehmer problem[J]. Electronic Research Archive, 2024, 32(3): 1864-1872. doi: 10.3934/era.2024085
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) $.
| [1] | R. K. Guy, Unsolved Problem in Number Theory, 3rd.edn, Springer-Verlag, New York, 2004. |
| [2] | W. P. Zhang, On a problem of D. H. Lehmer and its generalization, Compos. Math., 86 (1993), 307–316. |
| [3] | W. P. Zhang, A problem of D.H.Lehmer and its generalization (Ⅱ), Compos. Math., 91 (1994), 47–56. |
| [4] |
W. P. Zhang, On the difference between a D. H. Lehmer number and its inverse modulo $q$, Acta Arith., 68 (1994), 255–263. https://doi.org/10.4064/aa-68-3-255-263 doi: 10.4064/aa-68-3-255-263
|
| [5] | Y. N. Niu, R. Ma, H. D. Wang, On the difference between a D. H. Lehmer number and its inverse over short interval, arXiv preprint, (2021), arXiv: 2104.00216. https://doi.org/10.48550/arXiv.2104.00216 |
| [6] |
Y. M. Lu, Y. Yi, On the generalization of the D. H. Lehmer problem, Acta Math. Sin. (Engl. Ser.), 25 (2009), 1269–1274. https://doi.org/10.1007/s10114-009-7652-3 doi: 10.1007/s10114-009-7652-3
|
| [7] |
D. Han, Z. F. Xu, Y. Yi, T. P. Zhang, A Note on High-dimensional D. H. Lehmer Problem, Taiwanese J. Math., 25 (2021), 1137–1157. https://doi.org/10.11650/tjm/210705 doi: 10.11650/tjm/210705
|
| [8] |
Z. F. Xu, T. P. Zhang, High-dimensional D. H. Lehmer problem over short intervals, Acta Math. Sin. (Engl. Ser.), 30 (2014), 213–228. https://doi.org/10.1007/s10114-014-3324-z doi: 10.1007/s10114-014-3324-z
|
| [9] |
W. P. Zhang, On the distribution of primitive roots modulo $p$, Publ. Math. Debrecen, 53 (1998), 245–255. https://doi.org/10.5486/pmd.1998.1750 doi: 10.5486/pmd.1998.1750
|
| [10] |
C. I. Cobeli, S. M. Gonek, A. Zaharescu, On the distribution of small powers of a primitive root, J. Number Theory, 88 (2001), 49–58. https://doi.org/10.1006/jnth.2000.2604 doi: 10.1006/jnth.2000.2604
|
| [11] |
Z. Rudnick, A. Zaharescu. The distribution of spacings between small powers of a primitive root, Israel J. Math., 120 (2000), 271–287. https://doi.org/10.1007/s11856-000-1280-z doi: 10.1007/s11856-000-1280-z
|
| [12] |
I. E. Shparlinski, Distribution of exponential functions modulo a prime power, J. Number Theory, 143 (2014), 224–231. https://doi.org/10.1016/j.jnt.2014.04.010 doi: 10.1016/j.jnt.2014.04.010
|
Zhefeng Xu, Jiankang Wang, Lirong Zhu. On an exponential D. H. Lehmer problem[J]. Electronic Research Archive, 2024, 32(3): 1864-1872. doi: 10.3934/era.2024085
DownLoad: