On the distribution of the order over residue classes

Moree, Pieter

doi:10.1090/S1079-6762-06-00168-5

On the distribution of the order over residue classes

Author: Pieter Moree
Journal: Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 121-128
MSC (2000): Primary 11N37, 11R45; Secondary 11N69
DOI: https://doi.org/10.1090/S1079-6762-06-00168-5
Published electronically: August 18, 2006
MathSciNet review: 2263073
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Abstract: For a fixed rational number $g\not \in \{-1,0,1\}$ and integers $a$ and $d$ we consider the set $N_g(a,d)$ of primes $p$ such that the order of $g$ modulo $p$ is congruent to $a (\textrm {mod~}d)$. Under the Generalized Riemann Hypothesis (GRH), it can be shown that the set $N_g(a,d)$ has a natural density $\delta _g(a,d)$. Arithmetical properties of $\delta _g(a,d)$ are described, and $\delta _g(a,d)$ is compared with $\delta (a,d)$: the average density of elements in a field of prime characteristic having order congruent to $a (\textrm {mod~}d)$. It transpires that $\delta _g(a,d)$ has a strong tendency to be equal to $\delta (a,d)$, or at least to be close to it.

Koji Chinen and Leo Murata, On a distribution property of the residual order of $a\pmod p$. I, J. Number Theory 105 (2004), no. 1, 60–81. MR 2032442, DOI https://doi.org/10.1016/j.jnt.2003.06.005
Christopher Hooley, On Artin’s conjecture, J. Reine Angew. Math. 225 (1967), 209–220. MR 207630, DOI https://doi.org/10.1515/crll.1967.225.209
Pieter Moree, On the average number of elements in a finite field with order or index in a prescribed residue class, Finite Fields Appl. 10 (2004), no. 3, 438–463. MR 2067608, DOI https://doi.org/10.1016/j.ffa.2003.10.001
Pieter Moree, Convoluted convolved Fibonacci numbers, J. Integer Seq. 7 (2004), no. 2, Article 04.2.2, 14. MR 2084694
Pieter Moree, The formal series Witt transform, Discrete Math. 295 (2005), no. 1-3, 143–160. MR 2143453, DOI https://doi.org/10.1016/j.disc.2005.03.004

[M-0]M-old P. Moree, On primes $p$ for which $d$ divides ord$_p(g)$, Funct. Approx. Comment. Math. 33 (2005), 85–95.

Pieter Moree, On the distribution of the order and index of $g\pmod p$ over residue classes. I, J. Number Theory 114 (2005), no. 2, 238–271. MR 2167970, DOI https://doi.org/10.1016/j.jnt.2004.09.004
Pieter Moree, On the distribution of the order and index of $g\pmod p$ over residue classes. II, J. Number Theory 117 (2006), no. 2, 330–354. MR 2213769, DOI https://doi.org/10.1016/j.jnt.2005.06.006

[M-3]Moree3 P. Moree, On the distribution of the order and index of $g (\textrm {mod~}p)$ over residue classes III, arXiv:math.NT/0405527, Journal of Number Theory, to appear.

F. Pappalardi, On Hooley’s theorem with weights, Rend. Sem. Mat. Univ. Politec. Torino 53 (1995), no. 4, 375–388. Number theory, II (Rome, 1995). MR 1452393
K. Wiertelak, On the density of some sets of primes. IV, Acta Arith. 43 (1984), no. 2, 177–190. MR 736730, DOI https://doi.org/10.4064/aa-43-2-177-190
Kazimierz Wiertelak, On the density of some sets of primes $p$, for which $n\mid {\rm ord}_pa$, Funct. Approx. Comment. Math. 28 (2000), 237–241. Dedicated to Włodzimierz Staś on the occasion of his 75th birthday. MR 1824009, DOI https://doi.org/10.7169/facm/1538186700

[Z]Z D. Zagier, personal communication.