Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T10:28:38.065Z Has data issue: false hasContentIssue false
  • English
  • Français

An improvement of Artin's conjecture on average for composite moduli

Part of: Multiplicative number theory

Published online by Cambridge University Press:  26 February 2010

Shuguang Li
Shuguang Li
Affiliation:
Department of Mathematics, Natural Sciences Division, University of Hawaii at Hilo, 200 W. Kawili Street, Hilo. HI 96720–4091, USA. E-mail: [email protected]
Get access

Extract

Let q be a natural number. When the multiplicative iroup (ℤ/qℤ)* is a cyclic group, its generators are called primitive roots. Note that the generators are also elements with the maximum order if (ℤ/qℤ)* is cyclic. Thus, when (ℤ–qℤ)* is not a cyclic goup, we then call an element with: he maximal possible order a primitive root, which was initially introduced by R. Carmichael [1].

MSC classification

Secondary: 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004 , pp. 97 - 109
Copyright
Copyright © University College London 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Carmichael, R. D.. The Theory of Numbers. Wiley (New York, 1914).Google Scholar
2.Daverport, H.. Multiplicative Number Theory. Springer-Verlag (New York, 2000).Google Scholar
3.Gupta, R. and Murty, M. Ram. A remark on Artin's conjecture. Invent. Math., 78 (1984). 127–130.CrossRefGoogle Scholar
4.Heath-Brown, D. R.. Artin's conjecture for primitive roots. Quart. J. Math. Oxford (2), 37 (1986), 27–38.CrossRefGoogle Scholar
5.Hildebrand, A.. Large Values of Character Sums. J. Number Theory, 29 (1988), 271–296.CrossRefGoogle Scholar
6.Hooley, C.. On Artin's conjecture. J. reine angew. Math., 225 (1967), 209–220.Google Scholar
7.Ireland, K. and Rosen, M.. A Classical Introduction to Modern Number Theory (2nd ed. Springer-Verlag (New York, 1990).CrossRefGoogle Scholar
8.Li, S.. Artin's conjecture on average for composite moduli. J. Number Theory, 84 (2000), 93–118.Google Scholar
9.Li, S.. On extending Artin's conjecture to composite moduli. Mathematika, 46 (1999), 373–390.Google Scholar
10.Li, S. and Pomerance, C.. On generalizing Artin's conjecture on primitive roots to composite moduli. J. reine angew. Math., 556 (2003), 205–224.Google Scholar
11.Luca, F. and Pomerance, C.. On the average number of divisors of the Euler function. Publ. Math. Debrecen (to appear).Google Scholar
12.Murty, M. R.. Artin's conjecture for primitive roots. Math. Intelligencer, 10 (1988), no. 4, 59–67.Google Scholar
13.Polya, G.. Uber die Verteilung der quadratischen Reste und Nichtreste. Nachrkhten Königl. Ges. Wiss. Göttingen (1918), 30–36.Google Scholar
14.Stephens, P. J.. An average result for Artin's conjecture. Mathematika, 16 (1969), 178–188.Google Scholar
15.Tenenbaum, G.. Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press (Cambridge, 1995).Google Scholar
16.Vinogradov, I. M.. Uber die Verteilung der quadratischen Reste und Nichtreste. J. Soc. Phys. Math. Univ. Penni, 2 (1919), 1–14.Google Scholar