The correction factor in Artin's primitive root conjecture

Peter Stevenhagen

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 1, page 383-391
  • ISSN: 1246-7405

Abstract

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In 1927, E. Artin proposed a conjectural density for the set of primes p for which a given integer g is a primitive root modulo p . After computer calculations in 1957 by D. H. and E. Lehmer showed unexpected deviations, Artin introduced a correction factor to explain these discrepancies. The modified conjecture was proved by Hooley in 1967 under assumption of the generalized Riemann hypothesis. This paper discusses two recent developments with respect to the correction factor. The first is of historical nature, and is based on letters between Artin and the Lehmers from 1957-58 that were discovered in the Lehmer archives in Berkeley in december 2000. The second concerns a new interpretation of the correction factor in terms of local contributions by H. W. Lenstra, P. Moree and the author that is well-suited to deal with many generalizations of Artin’s original primitive root problem.

How to cite

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Stevenhagen, Peter. "The correction factor in Artin's primitive root conjecture." Journal de théorie des nombres de Bordeaux 15.1 (2003): 383-391. <http://eudml.org/doc/249101>.

@article{Stevenhagen2003,
abstract = {In 1927, E. Artin proposed a conjectural density for the set of primes $p$ for which a given integer $g$ is a primitive root modulo $p$. After computer calculations in 1957 by D. H. and E. Lehmer showed unexpected deviations, Artin introduced a correction factor to explain these discrepancies. The modified conjecture was proved by Hooley in 1967 under assumption of the generalized Riemann hypothesis. This paper discusses two recent developments with respect to the correction factor. The first is of historical nature, and is based on letters between Artin and the Lehmers from 1957-58 that were discovered in the Lehmer archives in Berkeley in december 2000. The second concerns a new interpretation of the correction factor in terms of local contributions by H. W. Lenstra, P. Moree and the author that is well-suited to deal with many generalizations of Artin’s original primitive root problem.},
author = {Stevenhagen, Peter},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {primitive roots; character; Frobenius element},
language = {eng},
number = {1},
pages = {383-391},
publisher = {Université Bordeaux I},
title = {The correction factor in Artin's primitive root conjecture},
url = {http://eudml.org/doc/249101},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Stevenhagen, Peter
TI - The correction factor in Artin's primitive root conjecture
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 1
SP - 383
EP - 391
AB - In 1927, E. Artin proposed a conjectural density for the set of primes $p$ for which a given integer $g$ is a primitive root modulo $p$. After computer calculations in 1957 by D. H. and E. Lehmer showed unexpected deviations, Artin introduced a correction factor to explain these discrepancies. The modified conjecture was proved by Hooley in 1967 under assumption of the generalized Riemann hypothesis. This paper discusses two recent developments with respect to the correction factor. The first is of historical nature, and is based on letters between Artin and the Lehmers from 1957-58 that were discovered in the Lehmer archives in Berkeley in december 2000. The second concerns a new interpretation of the correction factor in terms of local contributions by H. W. Lenstra, P. Moree and the author that is well-suited to deal with many generalizations of Artin’s original primitive root problem.
LA - eng
KW - primitive roots; character; Frobenius element
UR - http://eudml.org/doc/249101
ER -

References

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  1. [1] E. Artin, Collected papers. ed. S. Lang, J. T. Tate, Addison-Wesley, 1965. Zbl0493.01038MR176888
  2. [2] E. Artin, D.H. Lehmer, E. Lehmer, Correspondence 1957-58. Archives of D.H. Lehmer, Bancroft Library, Berkeley. 
  3. [3] H. Hasse, Vorlesungen über Zahlentheorie, Akademie-Verlag, 1950 MR51844
  4. [4] H. Hasse, Über die Artinsche Vermuting und verwandte Dichtefragen. Ann. Acad. Sci. Fennicae Ser. A. I. Math. Phys.116 (1952). Zbl0047.04204MR51864
  5. [5] C. Hooley, On Artin's conjecture. J. Reine Angew. Math.225 (1967), 209-220. Zbl0221.10048MR207630
  6. [6] D.H. Lehmer, E. Lehmer, Heuristics, anyone?. Studies in Mathematical Analysis and Related Topics, Stanford University Press, 1962. Zbl0129.02706MR144868
  7. [7] H.W. Lenstra, JR, On Artin's conjecture and Euclid's algorithm in global fields. Invent. Math.42 (1977), 201-224 Zbl0362.12012MR480413
  8. [8] H.W. Lenstra, JR, P. Moree, P. Stevenhagen, Character sums for primitive root densities, in preparation. Zbl06399690
  9. [9] P. Stevenhagen, H.W. Lenstra, JR, Chebotarëv and his density theorem. Math. Intellig.18 (1996), 26-37. Zbl0885.11005MR1395088

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