On Artin's Conjecture for Primitive Roots

Various generalizations of the Artin’s Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform version of the Chebotarev Density Theorem for the field Q(ζl, 2 ) valid for the range l < log x is proven. A uniform asymptotic formula for the number of primes up to x for which there exists a primitive root less than s is established. Lower bounds for the exponent of the class group of imaginary quadratic fields valid for density one sets of discriminants are determined. RESUME Nous considerons differentes generalisations de la conjecture d’Artin pour les racines primitives. Nous demontrons que pour au moins la moitie des nombres premiers p, les premiers log p nombres premiers engendrent une racine primitive. Nous demontrons une version uniforme du Theoreme de Densite de Chebotarev pour le corps Q(ζl, 2 ) pour l’intervalle l < log x. On etablit une formule asymptotique uniforme pour les nombres de premiers plus petits que x tels qu’ il existe une racine primitive plus petite que s. Nous determinons des minorants pour l’exposant du groupe de classe des corps quadratiques imaginaires valides pour ensembles de discriminants de densite 1.

[1]  K. R. Matthews.  A generalisation of Artin's conjecture for primitive roots , 1976 .

[2]  Leo Murata.  On the magnitude of the least prime primitive root , 1991 .

[3]  Gareth Jones,et al.  The Riemann Zeta-Function , 1992 .

[4]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[5]  Paul Erdös.  Über die kleinste quadratfreie Zahl einer arithmetischen Reihe , 1958 .

[6]  G. L..  Collected Papers , 1912, Nature.

[7]  de Ng Dick Bruijn.  The asymptotic behaviour of a function occuring in the theory of primes , 1951 .

[8]  M. Murty.  Artin’s conjecture for primitive roots , 1988 .

[9]  D. Hensley,et al.  The number of positive integers ≤x and free of prime factors >y , 1985 .

[10]  C. Hooley,et al.  Applications of Sieve Methods to the Theory of Numbers , 1976 .

[11]  Christopher Hooley.  A Note on Square‐Free Numbers in Arithmetic Progressions , 1975 .

[12]  M. Ram Murty,et al.  A remark on Artin's conjecture , 1984 .

[13]  W. Groß.  Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale , 1918 .

[14]  C. R. Matthews.  Counting Points Modulo p for some Finitely Generated Subgroups of Algebraic Groups , 1982 .

[15]  Cyclicity and generation of points mod p on elliptic curves , 1990 .

[16]  E. Landau.  Handbuch der Lehre von der Verteilung der Primzahlen , 1974 .

[17]  Paul Erdös,et al.  On the normal number of prime factors of p-1 and some related problems concerning euler's o/-function , 1935 .

[18]  E. Landau,et al.  Handbuch der Lehre von der Verteilung der Primzahlen , 1974 .

[19]  H. Davenport.  Multiplicative Number Theory , 1967 .

[20]  P Erdős,et al.  On the number of positive integers . . . , 1966 .

[21]  P Erdős At Budapest,et al.  On the Möbius function , .

[22]  M. Ram Murty,et al.  Cyclicity and generation of points modp on elliptic curves , 1990 .

[23]  A. Odlyzko,et al.  A bound for the least prime ideal in the Chebotarev Density Theorem , 1979 .

[24]  B. M. Fulk.  MATH , 1992 .

[25]  P. D. T. A. Elliott,et al.  The average of the least primitive root , 1968 .

[26]  On the exponent of the ideal class groups of complex quadratic fields. , 1972 .

[27]  S. Lang.  Algebraic Number Theory , 1971 .

[28]  G. Tenenbaum.  Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné , 1984 .

[29]  C. Hooley.  On Artin's conjecture. , 1967 .

[30]  H. Zassenhaus,et al.  Some empirical observations on primitive roots , 1971 .

[31]  de Ng Dick Bruijn.  On the number of positive integers $\leq x$ and free of prime factors $>y$ , 1951 .

[32]  An analogue of Artin's conjecture for Abelian extensions , 1984 .

[33]  M. V. Ramana Murty,et al.  Primitive points on elliptic curves , 1986 .

[34]  V. Murty.  Explicit formulae and the Lang-Trotter conjecture , 1985 .

[35]  P. Erdös,et al.  On a problem of Oppenheim concerning “factorisatio numerorum” , 1983 .

[36]  Jean-Pierre Serre,et al.  Linear representations of finite groups , 1977, Graduate texts in mathematics.