On the $r$-rank Artin Conjecture
HTML articles powered by AMS MathViewer
- by Francesco Pappalardi PDF
- Math. Comp. 66 (1997), 853-868 Request permission
Abstract:
We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which $\mathbb {F}_p^*$ can be generated by $r$ given multiplicatively independent numbers. In the case when the $r$ given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown–Zassenhaus (J. Number Theory 3 (1971), 306–309). Finally, in some examples, we compare the densities approximated with the natural densities calculated with primes up to $9\cdot 10^4$.References
- K. A. Hirsch, On skew-groups, Proc. London Math. Soc. 45 (1939), 357–368. MR 0000036, DOI 10.1112/plms/s2-45.1.357
- H. Brown and H. Zassenhaus, Some empirical observations on primitive roots, J. Number Theory 3 (1971), 306–309. MR 288072, DOI 10.1016/0022-314X(71)90004-7
- P. D. T. A. Elliott, A problem of Erdős concerning power residue sums, Acta Arith. 13 (1967/68), 131–149. MR 220689, DOI 10.4064/aa-13-2-131-149
- P. D. T. A. Elliott, Corrigendum: “A problem of Erdős concerning power residue sums”, Acta Arith. 14 (1967/68), 437. MR 228451
- S. W. Graham and C. J. Ringrose, Lower bounds for least quadratic nonresidues, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 269–309. MR 1084186
- Rajiv Gupta and M. Ram Murty, Primitive points on elliptic curves, Compositio Math. 58 (1986), no. 1, 13–44. MR 834046
- Christopher Hooley, On Artin’s conjecture, J. Reine Angew. Math. 225 (1967), 209–220. MR 207630, DOI 10.1515/crll.1967.225.209
- J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 409–464. MR 0447191
- C. R. Matthews, Counting points modulo $p$ for some finitely generated subgroups of algebraic groups, Bull. London Math. Soc. 14 (1982), no. 2, 149–154. MR 647199, DOI 10.1112/blms/14.2.149
- A. I. Vinogradov, Artin’s $L$-series and his conjectures, Trudy Mat. Inst. Steklov. 122 (1971), 123–140, 387 (Russian). Collection of articles dedicated to Academician Ivan Matveevič Vinogradov on his eightieth birthday, I. MR 0340222
Additional Information
- Francesco Pappalardi
- Affiliation: Dipartimento di Matematica, Università degli Studi di Roma Tre, Via C. Segre, 2, 00146 Rome, Italy
- Email: pappa@matrm3.mat.uniroma3.it
- Received by editor(s): April 11, 1995
- Received by editor(s) in revised form: January 23, 1996
- Additional Notes: Supported by Human Capital and Mobility Program of the European Community, under contract ERBCHBICT930706
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 853-868
- MSC (1991): Primary 11N37; Secondary 11N56
- DOI: https://doi.org/10.1090/S0025-5718-97-00805-3
- MathSciNet review: 1377664