Short effective intervals containing primes in arithmetic progressions and the seven cubes problem
HTML articles powered by AMS MathViewer
- by H. Kadiri PDF
- Math. Comp. 77 (2008), 1733-1748 Request permission
Abstract:
For any $\epsilon >0$ and any non-exceptional modulus $q\ge 3$, we prove that, for $x$ large enough ($x\ge \alpha _{\epsilon }\log ^2 q$), the interval $\left [ e^x,e^{x+\epsilon }\right ]$ contains a prime $p$ in any of the arithmetic progressions modulo $q$. We apply this result to establish that every integer $n$ larger than $\exp (71 000)$ is a sum of seven cubes.References
- Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
- D. R. Heath-Brown, Zero-free regions for Dirichlet $L$-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), no. 2, 265–338. MR 1143227, DOI 10.1112/plms/s3-64.2.265
- Habiba Kadiri, Une région explicite sans zéros pour la fonction $\zeta$ de Riemann, Acta Arith. 117 (2005), no. 4, 303–339 (French). MR 2140161, DOI 10.4064/aa117-4-1
- H. Kadiri, An explicit zero-free region for Dirichlet $L$-functions. submitted, can be found at http://arxiv.org/pdf/math.NT/0510570.
- U. V. Linnik, On the representation of large numbers as sums of seven cubes, Rec. Math. [Mat. Sbornik] N. S. 12(54) (1943), 218–224 (English, with Russian summary). MR 0009388
- Ming-Chit Liu and Tianze Wang, Distribution of zeros of Dirichlet $L$-functions and an explicit formula for $\psi (t,\chi )$, Acta Arith. 102 (2002), no. 3, 261–293. MR 1884719, DOI 10.4064/aa102-3-5
- Kevin S. McCurley, Explicit estimates for the error term in the prime number theorem for arithmetic progressions, Math. Comp. 42 (1984), no. 165, 265–285. MR 726004, DOI 10.1090/S0025-5718-1984-0726004-6
- Kevin S. McCurley, An effective seven cube theorem, J. Number Theory 19 (1984), no. 2, 176–183. MR 762766, DOI 10.1016/0022-314X(84)90100-8
- Olivier Ramaré and Robert Rumely, Primes in arithmetic progressions, Math. Comp. 65 (1996), no. 213, 397–425. MR 1320898, DOI 10.1090/S0025-5718-96-00669-2
- Olivier Ramaré and Yannick Saouter, Short effective intervals containing primes, J. Number Theory 98 (2003), no. 1, 10–33. MR 1950435, DOI 10.1016/S0022-314X(02)00029-X
- O. Ramaré, An explicit seven cube theorem, Acta Arith. 118 (2005), no. 4, 375–382. MR 2165551, DOI 10.4064/aa118-4-4
- Barkley Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211–232. MR 3018, DOI 10.2307/2371291
- J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
- Samuel S. Wagstaff Jr., Greatest of the least primes in arithmetic progressions having a given modulus, Math. Comp. 33 (1979), no. 147, 1073–1080. MR 528061, DOI 10.1090/S0025-5718-1979-0528061-7
- G. L. Watson, A proof of the seven cube theorem, J. London Math. Soc. 26 (1951), 153–156. MR 47691, DOI 10.1112/jlms/s1-26.2.153
Additional Information
- H. Kadiri
- Affiliation: Département de Mathématiques et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC H3C 3J7, Canada
- Address at time of publication: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, Canada T1K 3M4
- MR Author ID: 760548
- Email: habiba.kadiri@uleth.ca
- Received by editor(s): August 29, 2006
- Received by editor(s) in revised form: July 7, 2007
- Published electronically: February 8, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 1733-1748
- MSC (2000): Primary 11M26
- DOI: https://doi.org/10.1090/S0025-5718-08-02084-X
- MathSciNet review: 2398791