Conditional bounds for the least quadratic non-residue and related problems. (English) Zbl 1326.11058
Math. Comput. 84, No. 295, 2391-2412 (2015); corrigendum ibid. 86, No. 307, 2551-2554 (2017).
Let \(q>1\) be a positive number and let \(G = (\mathbb Z/q\mathbb Z)^*\) denote the group of reduced residues \(\pmod{q}\) and let \(H\) be a proper subgroup of \(G\). Assuming the generalized Riemann hypothesis the authors prove:
- (1)
- If \(q \geq 3000\) is an integer then the least prime \(\ell\) with \(\ell\not |q\) and \(\ell\) not lying in the subgroup \(H\) satisfies the bound \(\ell\leq (\log q+B(q))^2\) where \(B(q)\) is explicitly given.
- (2)
- If the index \(h=|G:H|\geq4\) then the least prime \(p\) not in \(H\) satisfies \(p<(\alpha(h)+o(1))(\log q)^2\) again with \(\alpha(h)\) explicitly given.
- (3)
- If \(q\geq 20000\) and \(h>1\) then the smallest prime \(p\) lying in a given coset \(aH\) is either \(\leq 10^9\) or \(\leq\left((h-1)\log q+3(h+1)+(5/2)(\log\log q)^2\right)^2\) (the proof gives actually a more precise estimate).
- (4)
- New explicit upper and lower bounds for \(L\)-values at the edge of the critical strip and explicit bounds for the class number of imaginary quadratic fields.
Reviewer: Štefan Porubský (Praha)
MSC:
| 11N60 | Distribution functions associated with additive and positive multiplicative functions |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
Keywords:
generalized Riemann hypothesis; least quadratic non-residue; least \(k\)-th power non-residue; least prime in an arithmetic progression; Littlewood bounds for \(L\)-functions; lower bounds for \(L(1,\chi)\) and \(\zeta(1 + it)\); bounds for the class number of imaginary quadratic fieldsReferences:
| [1] | Ankeny, N. C., The least quadratic non residue, Ann. of Math. (2), 55, 65-72 (1952) · Zbl 0046.04006 |
| [2] | Bach, Eric, Explicit bounds for primality testing and related problems, Math. Comp., 55, 191, 355-380 (1990) · Zbl 0701.11075 · doi:10.2307/2008811 |
| [3] | Booker, Andrew R., Quadratic class numbers and character sums, Math. Comp., 75, 255, 1481-1492 (electronic) (2006) · Zbl 1095.11057 · doi:10.1090/S0025-5718-06-01850-3 |
| [4] | Bach, Eric; Sorenson, Jonathan, Explicit bounds for primes in residue classes, Math. Comp., 65, 216, 1717-1735 (1996) · Zbl 0853.11077 · doi:10.1090/S0025-5718-96-00763-6 |
| [5] | Carneiro, Emanuel; Chandee, Vorrapan, Bounding \(\zeta (s)\) in the critical strip, J. Number Theory, 131, 3, 363-384 (2011) · Zbl 1231.11097 · doi:10.1016/j.jnt.2010.08.002 |
| [6] | Chandee, Vorrapan, Explicit upper bounds for \(L\)-functions on the critical line, Proc. Amer. Math. Soc., 137, 12, 4049-4063 (2009) · Zbl 1243.11088 · doi:10.1090/S0002-9939-09-10075-8 |
| [7] | Chandee, Vorrapan; Soundararajan, K., Bounding \(\vert \zeta (\frac 12+it)\vert\) on the Riemann hypothesis, Bull. Lond. Math. Soc., 43, 2, 243-250 (2011) · Zbl 1238.11078 · doi:10.1112/blms/bdq095 |
| [8] | Davenport, Harold, Multiplicative Number Theory, Graduate Texts in Mathematics 74, xiv+177 pp. (2000), Springer-Verlag: New York:Springer-Verlag · Zbl 1002.11001 |
| [9] | Hafner, James L.; McCurley, Kevin S., A rigorous subexponential algorithm for computation of class groups, J. Amer. Math. Soc., 2, 4, 837-850 (1989) · Zbl 0702.11088 · doi:10.2307/1990896 |
| [10] | Heath-Brown, D. R., Zero-free regions for Dirichlet \(L\)-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3), 64, 2, 265-338 (1992) · doi:10.1112/plms/s3-64.2.265 |
| [11] | Iwaniec, Henryk; Kowalski, Emmanuel, Analytic Number Theory, American Mathematical Society Colloquium Publications 53, xii+615 pp. (2004), American Mathematical Society: Providence, RI:American Mathematical Society · Zbl 1059.11001 |
| [12] | Jacobson, Michael J., Jr.; Ramachandran, Shantha; Williams, Hugh C., Numerical results on class groups of imaginary quadratic fields. Algorithmic number theory, Lecture Notes in Comput. Sci. 4076, 87-101 (2006), Springer: Berlin:Springer · Zbl 1143.11370 · doi:10.1007/11792086\_7 |
| [13] | [Li1] J. E. Littlewood, On the zeros of the Riemann zeta-function, Math. Proc. Cambridge Philos. Soc. 22 (1924) 295-318. · JFM 50.0230.02 |
| [14] | [Li2] J. E. Littlewood, On the function \(1/\zeta(1+it)\), Proc. London Math. Soc. 27 (1928) 349-357. · JFM 54.0367.03 |
| [15] | Watkins, Mark, Class numbers of imaginary quadratic fields, Math. Comp., 73, 246, 907-938 (electronic) (2004) · Zbl 1050.11095 · doi:10.1090/S0025-5718-03-01517-5 |
| [16] | [Wed] S. Wedeniwski, Primality Tests on Commutator Curves. Dissertation: T\"ubingen, Germany, 2001. |
| [17] | Xylouris, Triantafyllos, On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet \(L\)-functions, Acta Arith., 150, 1, 65-91 (2011) · Zbl 1248.11067 · doi:10.4064/aa150-1-4 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
