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Artin's conjecture on the average

Published online by Cambridge University Press:  26 February 2010

Morris Goldfeld
Morris Goldfeld
Affiliation:
Columbia University, New York 27, N.Y.
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Extract

It was conjectured by Artin [1] that each non-zero integer a unequal to +1, −1 or a perfect square is a primitive root for infinitely many primes p. More precisely, denoting by Na(x) the number of primes p ≤ x for which a is a primitive root, he conjectured that

where c(a) is a positive constant. This conjecture has recently been proved by C. Hooley [2] under the assumption that the Riemann hypothesis holds for fields of the type .

Type
Research Article
Information
Mathematika , Volume 15 , Issue 2 , December 1968 , pp. 223 - 226
Copyright
Copyright © University College London 1968

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References

1.Lang, S. and Tate, J., The Collected Papers of Emil Artin (Addison-Wesley, 1965), Preface.Google Scholar
2.Hooley, C., “On Artin's Conjecture”, Journal fur Math., 225 (1967), 209–220.Google Scholar
3.Prachar, K., Primzahlverteilung (Springer, 1957).Google Scholar
4.Heilbronn, H., “On the averages of some arithmetical functions of two variables”, Mathematika, 5 (1958), 1–7.CrossRefGoogle Scholar
5.Gallagher, P. X., “The large sieve”, Mathematika, 14 (1967), 14–20.CrossRefGoogle Scholar
6.Vinogradov, I. M., The method of trigonometric sums in the theory of numbers (Interscience, 1961), 41–43.Google Scholar