The least inert prime in a real quadratic field

Treviño, Enrique

doi:10.1090/S0025-5718-2012-02579-8

The least inert prime in a real quadratic field

Author: Enrique Treviño
Journal: Math. Comp. 81 (2012), 1777-1797
MSC (2010): Primary 11L40, 11Y40, 11R11
DOI: https://doi.org/10.1090/S0025-5718-2012-02579-8
Published electronically: February 10, 2012
MathSciNet review: 2904602
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Abstract: In this paper, we prove that for any positive fundamental discriminant , there is always at least one prime $p \leq D^{0.45}$ such that the Kronecker symbol . This improves a result of Granville, Mollin and Williams, where they showed that the least inert prime in a real quadratic field of discriminant is at most $\sqrt {D}/2$ . We use a ``smoothed'' version of the Pólya-Vinogradov inequality, which is very useful for numerically explicit estimates.

1. Eric Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355–380. MR 1023756, https://doi.org/10.1090/S0025-5718-1990-1023756-8
2. D. A. Burgess, On character sums and primitive roots, Proc. London Math. Soc. (3) 12 (1962), 179–192. MR 0132732, https://doi.org/10.1112/plms/s3-12.1.179
3. Richard Crandall and Carl Pomerance, Prime numbers, 2nd ed., Springer, New York, 2005. A computational perspective. MR 2156291
4. P. Dusart.
Sharper bounds for $\psi$ , $\theta$ , $\pi$ ,.
Laboratoire d'Arithmétique, de Calcul formel et d'Optimisation, 1998.
5. Pierre Dusart, Inégalités explicites pour 𝜓(𝑋), 𝜃(𝑋), 𝜋(𝑋) et les nombres premiers, C. R. Math. Acad. Sci. Soc. R. Can. 21 (1999), no. 2, 53–59 (French, with English and French summaries). MR 1697455
6. Andrew Granville, R. A. Mollin, and H. C. Williams, An upper bound on the least inert prime in a real quadratic field, Canad. J. Math. 52 (2000), no. 2, 369–380. MR 1755783, https://doi.org/10.4153/CJM-2000-017-5
7. M. Levin, C. Pomerance, and K. Soundararajan.
Fixed points for discrete logarithms.
In Algorithmic number theory, volume 6197 of Lecture Notes in Comput. Sci., pages 6-15. 2010.
8. R. F. Lukes, C. D. Patterson, and H. C. Williams, Some results on pseudosquares, Math. Comp. 65 (1996), no. 213, 361–372, S25–S27. MR 1322892, https://doi.org/10.1090/S0025-5718-96-00678-3
9. Richard A. Mollin, Quadratics, CRC Press Series on Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 1996. MR 1383823
10. Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
11. Karl K. Norton, Numbers with small prime factors, and the least 𝑘th power non-residue, Memoirs of the American Mathematical Society, No. 106, American Mathematical Society, Providence, R.I., 1971. MR 0286739
12. Karl K. Norton, Bounds for sequences of consecutive power residues. I, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 213–220. MR 0332697
13. J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689
14. Lowell Schoenfeld, Sharper bounds for the Chebyshev functions 𝜃(𝑥) and 𝜓(𝑥). II, Math. Comp. 30 (1976), no. 134, 337–360. MR 0457374, https://doi.org/10.1090/S0025-5718-1976-0457374-X
15. Jonathan P. Sorenson, Sieving for pseudosquares and pseudocubes in parallel using doubly-focused enumeration and wheel datastructures, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 6197, Springer, Berlin, 2010, pp. 331–339. MR 2721430, https://doi.org/10.1007/978-3-642-14518-6_26
16. E. Treviño.
Numerically explicit estimates for character sums.
2011.
Thesis (Ph.D.), Dartmouth College.
17. K. Wooding.
The sieve problem in one- and two-dimensions.
2010.
Thesis (Ph.D.), University of Calgary.