You are here

The Generalized Artin Primitive Root Conjecture

Journal Name:

Publication Year:

Author Name
Abstract (2. Language): 
Asymptotic formulas for the number of integers with the primitive root 2, and the generalized Artin conjecture for subsets of composite integers with xed admissible primitive roots u 6= 1; v2, are presented here.
23
34

JEL Codes:

REFERENCES

References: 

[1] Apostol, Tom M. Introduction to analytic number theory. Undergraduate Texts in
Mathematics. Springer-Verlag, New York-Heidelberg, 1976.
[2] Balog, Antal; Cojocaru, Alina-Carmen; David, Chantal. Average twin prime conjecture
for elliptic curves. Amer. J. Math. 133,(2011), no. 5, 1179-1229.
[3] Roger C. Baker, Paul Pollack, Bounded gaps between primes with a given primitive
root, II, arXiv:1407.7186.
[4] Carmichael, R. D. Note on a new number theory function. Bull. Amer. Math. Soc.
16 (1910), no. 5, 232-238.
[5] Peter J. Cameron and D. A. Preece, Notes on primitive lambda-roots,
http://www.maths.qmul.ac.uk/~pjc/csgnotes/lambda.pdf
[6] Joseph Cohen, Primitive roots in quadratic elds, II, Journal of Number Theory 124
(2007) 429-441.
[7] H. Davenport, On Primitive Roots in Finite Fields, Quarterly J. Math. 1937, 308-
312.
[8] Rainer Dietmann, Christian Elsholtz, Igor E. Shparlinski, On Gaps Between Primitive
Roots in the Hamming Metric, arXiv:1207.0842.
[9] Paul Erdos, Harold N. Shapiro, On The Least Primitive Root Of A Prime, 1957,
euclidproject.org.
REFERENCES 33
[10] Gupta, Rajiv; Murty, M. Ram. A remark on Artin's conjecture. Invent. Math. 78
(1984), no. 1, 127-130.
[11] Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Sixth edition.
Revised by D. R. Heath-Brown and J. H. Silverman. With a foreword by Andrew
Wiles. Oxford University Press, 2008.
[12] Hildebrand, Adolf. Quantitative mean value theorems for nonnegative multiplicative
functions. II. Acta Arith. 48 (1987), no. 3, 209-260.
[13] C. Hooley, On Artins conjecture, J. Reine Angew. Math. 225, 209-220, 1967.
[14] Li, Shuguang; Pomerance, Carl. Primitive roots: a survey. Number theoretic methods,
Iizuka, 2001, 219-231, Dev. Math., 8, Kluwer Acad. Publ., Dordrecht, 2002.
[15] Li, Shuguang; Pomerance, Carl. On generalizing Artin's conjecture on primitive
roots to composite moduli. J. Reine Angew. Math. 556 (2003), 205-224.
[16] Iwaniec, Henryk; Kowalski, Emmanuel. Analytic number theory. AMS Colloquium
Publications, 53. American Mathematical Society, Providence, RI, 2004.
[17] Konyagin, Sergei V.; Shparlinski, Igor E. On the consecutive powers of a primitive
root: gaps and exponential sums. Mathematika 58 (2012), no. 1, 11-20.
[18] H. W. Lenstra Jr, P. Moree, P. Stevenhagen, Character sums for primitive root
densities, arXiv:1112.4816.
[19] Lenstra, H. W., Jr. On Artin conjecture and Euclid algorithm in global elds. Invent.
Math. 42, (1977), 201-224.
[20] Lidl, Rudolf; Niederreiter, Harald. Finite elds. With a foreword by P. M. Cohn.
Second edition. Encyclopedia of Mathematics and its Applications, 20. Cambridge
University Press, Cambridge, 1997.
[21] Moree, Pieter. Counting numbers in multiplicative sets: Landau versus Ramanujan.
Math. Newsl. 21 (2011), no. 3, 73-81.
[22] Pieter Moree. Artin's primitive root conjecture -a survey. arXiv:math/0412262.
[23] Montgomery, Hugh L.; Vaughan, Robert C. Multiplicative number theory. I. Classical
theory. Cambridge University Press, Cambridge, 2007.
[24] Narkiewicz, W. The development of prime number theory. From Euclid to Hardy and
Littlewood. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000.
[25] Pappalardi, Francesco; Saidak, Filip; Shparlinski, Igor E. Square-free values of the
Carmichael function. J. Number Theory 103 (2003), no. 1, 122-131.
REFERENCES 34
[26] Pappalardi, Francesco; Susa, Andrea. An analogue of Artin conjecture for multiplicative
subgroups of the rationals. Arch. Math. (Basel) 101, (2013), no. 4, 319-330.
[27] A. G. Postnikov, Introduction to analytic number theory, Translations of Mathematical
Monographs, vol. 68, American Mathematical Society, Providence, RI, 1988.
[28] Paszkiewicz, A. A new prime p for which the least primitive root mod p and the least
primitive root modp2 are not equal. Math. Comp. 78 (2009), no. 266, 1193-1195.
[29] Roskam, Hans. Artin primitive root conjecture for quadratic elds. J. Theory Nombres
Bordeaux, 14, (2002), no. 1, 287-324.
[30] Rose, H. E. A course in number theory. Second edition. Oxford Science Publications.
The Clarendon Press, Oxford University Press, New York, 1994.
[31] Ribenboim, Paulo, The new book of prime number records, Berlin, New York:
Springer-Verlag, 1996.
[32] Stevenhagen, Peter. The correction factor in Artin's primitive root conjecture. Les
XXII emes Journees Arithmetiques (Lille, 2001). J. Theor. Nombres Bordeaux 15
(2003), no. 1, 383-391.
[33] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge
Studies in Advanced Mathematics 46, Cambridge University Press, Cambridge, 1995.
[34] Stephens, P. J. An average result for Artin conjecture. Mathematika 16, (1969),
178-188.
[35] Vaughan, R. C. Some applications of Montgomery's sieve. J. Number Theory 5
(1973), 64-79.
[36] E. Wirsing, Das asymptotische Verhalten von Summen uber multiplikative Funktionen,
Math. Ann. 143 (1961) 75-102.
[37] Williams, Kenneth S. Note on integers representable by binary quadratic forms.
Canad. Math. Bull. 18 (1975), no. 1, 123-125.

Thank you for copying data from http://www.arastirmax.com