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The Distribution of Primitive Roots

Published online by Cambridge University Press:  20 November 2018

P. D. T. A. Elliott
P. D. T. A. Elliott*
Affiliation:
University of Nottingham, Nottingham, England
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Notation,p and q are generic symbols for prime numbers. N(H, p) denotes the number of primes q, not exceeding Hy which are primitive roots (mod p).

g(p) denotes the least positive primitive root (mod p).

g*(p) is the least prime primitive root (mod p).

v(m) denotes the number of distinct prime divisors of the integer m.

τk (m) is the number of ways of representing the integer m as the product of k integers, order being important.

π(x, k, r) is the number of primes p, not exceeding x, which satisfy p ≡ r(mod k); while π(x) denotes the total number of p ≦ x.

logm x denotes the mth iterated logarithmic function which is defined, when possible, by

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 822 - 841
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Ankeny, N. C., The least quadratic non-residue, Ann. of Math. (2) 55 (1952), 65–71.Google Scholar
2. Artin, E., Collected papers (Addison-Wesley, Reading, Massachusetts, 1965).Google Scholar
3. Barban, M. B., On a theorem of I. P. Kubilius, Izv. Akad. Nauk USSR Ser. Fiz.-Mat. Nauk 1961 (5), 3–9. (Russian)Google Scholar
4. Bombieri, E., On the large sieve, Mathematika 12 (1965), 201–225.Google Scholar
5. Burgess, D. A., On character sums and primitive roots, Proc. London Math. Soc. (3) 12 (1962), 179–192.Google Scholar
6. Burgess, D. A. and Elliott, P. D. T. A., On the average value of the least primitive root, Mathematika 15 (1968), 39–50.Google Scholar
7. Davenport, H. and Erdos, P., The distribution of quadratic and higher residues, Publ. Math. (Debrecen) 2 (1952), 252–265.Google Scholar
8. Elliott, P. D. T. A., On the size of L(l, x ), J. Reine Angew. Math. 236 (1969), 26–36 Google Scholar
9. Elliott, P. D. T. A., On the mean value of f(p) (to appear).Google Scholar
10. Fogels, E., On the distribution of prime ideals, Acta Arithmetica VII (1962), 255–269.Google Scholar
11. Gallagher, P. X., The large sieve, Mathematika 14 (1967), 14–20.Google Scholar
12. Gallagher, P. X., Bombieri1 s mean-value theorem, Mathematika 15 (1968), 1–6.Google Scholar
13. Halberstam, H. and Richert, H. E., Sieve methods (to be published by Markham, Chicago).Google Scholar
14. Halberstam, H. and Roth, K. F., Sequences, Vol. 1 (Oxford Univ. Press, Oxford, 1968).Google Scholar
15. Halberstam, H., Jurkat, W., and Richert, H. E., Un nouveau résultat de la méthode du crible, C. R. Acad. Sci. Paris 264 (1967), 920–923 Google Scholar
16. Hooley, C., On Artin1 s conjecture, J. Reine Angew. Math. 225 (1967), 209–220.Google Scholar
17. Kubilius, I. P., Probabilistic methods in the theory of numbers, Amer. Math. Soc. Transi. Vol. 11 (Amer. Math. Soc, Providence, R. I., 1964).Google Scholar
18. Landau, E., Vorlesungen ilber Zahlentheorie, Band 2, Teil VII, Kap. 14 (Leipzig, Berlin, 1927).Google Scholar
19. Linnik, U. V., On the least prime in arithmetic progression. I. The basic theorem; II. The Deuring-Heilbronn phenomenon, Mat. Sb. (N.S.) 15 (57) (1944), 139-178; 347–367.Google Scholar
20. Prachar, K., Primzahlverteilung, (Berlin, 1957).Google Scholar
21. Rényi, A., On the representation of an even number as the sum of a prime and an almost prime, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 57–78 Google Scholar
22. Wang, Y., On the least primitive root of a prime, Sci. Sinica X (1) (1961), 1–14.Google Scholar