Skip to main content
SpringerLink
Log in

The distribution of the multiplicative index of algebraic numbers over residue classes

  • Published:
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Aims and scope Submit manuscript

Abstract

Let K be a number field and G a finitely generated torsion-free subgroup of \(K^\times \). Given a prime \(\mathfrak {p}\) of K we denote by \({{\,\textrm{ind}\,}}_\mathfrak {p}(G)\) the index of the subgroup \((G\bmod \mathfrak {p})\) of the multiplicative group of the residue field at \(\mathfrak {p}\). Under the Generalized Riemann Hypothesis we determine the natural density of primes of K for which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extension F of K. We study in detail the natural density in case S is an arithmetic progression, in particular its positivity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrose, C.: Artin’s primitive root conjecture and a problem of Rohrlich. Math. Proc. Cambridge Philos. Soc. 157(1), 79–99 (2014)

    Article  MathSciNet  Google Scholar 

  2. Chinen, K., Murata, L.: On a distribution property of the residual order of \(a\,(\text{ mod }~p)\), I, II. J. Number Theory 105, 60–81 (2004)

    Article  MathSciNet  Google Scholar 

  3. Felix, A.T.: Higher rank generalizations of Fomenko’s conjecture. J. Number Theory 133(5), 1738–1751 (2013)

    Article  MathSciNet  Google Scholar 

  4. Felix, A.T.: The index of \(a\) modulo \(p\), SCHOLAR-a scientific celebration highlighting open lines of arithmetic research. Contemp. Math. 655, 83–96 (2015)

    Article  Google Scholar 

  5. Felix, A.T., Murty, M.R.: A problem of Fomenko’s related to Artin’s conjecture. Int. J. Number Theory 8(7), 1687–1723 (2012)

    Article  MathSciNet  Google Scholar 

  6. Gupta, R., Murty, M.R.: A remark on Artin’s conjecture. Invent. Math. 78, 127–130 (1984)

    Article  MathSciNet  Google Scholar 

  7. Heath-Brown, D.R.: Artin’s conjecture for primitive roots. Quart. J. Math. Oxford Ser. 37, 27–38 (1986)

    Article  MathSciNet  Google Scholar 

  8. Hooley, C.: On Artin’s conjecture. J. Reine Angew. Math. 225, 209–220 (1967)

    MathSciNet  Google Scholar 

  9. Järviniemi, O., Perucca, A.: Unified treatment of Artin-type problems. Res. Number Theory 9(10), 10 (2023). https://doi.org/10.1007/s40993-022-00418-6

    Article  MathSciNet  Google Scholar 

  10. Landau, E.: Über die zahlentheoretische Function \(\varphi (n)\) und ihre Beziehung zum Goldbachschen Satz, Göttinger Nachrichten, 177–186. Collected Works I, 106–115 (1900)

    Google Scholar 

  11. Moree, P.: On the average number of elements in a finite field with order or index in a prescribed residue class. Finite Fields Appl. 10(3), 438–463 (2004)

    Article  MathSciNet  Google Scholar 

  12. Moree, P.: On the distribution of the order and index of \(g\,(\text{ mod }~p)\) over residue classes - I. J. Number Theory 114(2), 238–271 (2005)

    Article  MathSciNet  Google Scholar 

  13. Moree, P.: On the distribution of the order and index of \(g\,(\text{ mod }~p)\) over residue classes - II. J. Number Theory 117(2), 330–354 (2006)

    Article  MathSciNet  Google Scholar 

  14. Moree, P.: On primes \(p\) for which \(d\) divides \(\text{ ord}_p (g)\). Funct. Approx. Comment. Math. 33, 85–95 (2005)

    Article  MathSciNet  Google Scholar 

  15. Moree, P.: On the distribution of the order over residue classes. Electron. Res. Announc. Am. Math. Soc. 12, 121–128 (2006)

    Article  MathSciNet  Google Scholar 

  16. Moree, P., Sha, M.: Primes in arithmetic progressions and nonprimitive roots. Bull. Aust. Math. Soc. 100(3), 388–394 (2019)

    Article  MathSciNet  Google Scholar 

  17. Pappalardi, F.: On Hooley’s theorem with weights. Rend. Sem. Mat. Univ. Pol. Torino 53(4), 375–388 (1995)

    MathSciNet  Google Scholar 

  18. Pappalardi, F.: On the \(r\)-rank Artin conjecture. Math. Comp. 66(218), 853–868 (1997)

    Article  MathSciNet  Google Scholar 

  19. Perucca, A., Sgobba, P.: Kummer theory for number fields and the reductions of algebraic numbers. Int. J. Number Theory 15(08), 1617–1633 (2019)

    Article  MathSciNet  Google Scholar 

  20. Perucca, A., Sgobba, P.: Kummer theory for number fields and the reductions of algebraic numbers II. Uniform Distrib. Theory 15(1), 75–92 (2020)

    Article  MathSciNet  Google Scholar 

  21. Perucca, A., Sgobba, P., Tronto, S.: The degree of Kummer extensions of number fields. Int. J. Number Theory 17(5), 1091–1110 (2021)

    Article  MathSciNet  Google Scholar 

  22. Wiertelak, K.: On the density of some sets of primes - IV. Acta Arith. 43(2), 177–190 (1984)

    Article  MathSciNet  Google Scholar 

  23. Wiertelak, K.: On the density of some sets of primes \(p\), for which \(n|\text{ ord}_p(a)\). Funct. Approx. Comment. Math. 28, 237–241 (2000)

    Article  MathSciNet  Google Scholar 

  24. Ziegler, V.: On the distribution of the order of number field elements modulo prime ideals. Uniform Distrib. Theory 1(1), 65–85 (2006)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This project was started when the first author gave a talk in the Luxembourg Number Theory Day 2019. He thanks the other authors for the invitation and for several subsequent invitations. The second and third author thank the Max Planck Institut für Mathematik and the first author for organizing a short visit in October 2022. Thanks are also due to Alessandro Languasco for verifying Table 2 using Theorem 16. We thank Valentin Blomer for pointing out reference [1].

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Pieter Moree

  2. Department of Mathematics, University of Luxembourg, 6 Avenue de la Fonte, L-4364, Esch-sur-Alzette, Luxembourg

    Antonella Perucca

  3. Department of Pure Mathematics, Xi’an Jiaotong–Liverpool University, 111 Ren’ai Road, Suzhou, 215123, China

    Pietro Sgobba

Authors
  1. Pieter Moree
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antonella Perucca
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pietro Sgobba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pietro Sgobba.

Ethics declarations

Conflicts of interest

The authors have no conflicts of interest to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moree, P., Perucca, A. & Sgobba, P. The distribution of the multiplicative index of algebraic numbers over residue classes. Abh. Math. Semin. Univ. Hambg. (2024). https://doi.org/10.1007/s12188-024-00276-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12188-024-00276-2

Keywords

Mathematics Subject Classification

Search

Navigation