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Primes in arithmetic progressions


Authors: Olivier Ramaré and Robert Rumely
Journal: Math. Comp. 65 (1996), 397-425
MSC (1991): Primary 11N13, 11N56, 11M26; Secondary 11Y35, 11Y40, 11--04
DOI: https://doi.org/10.1090/S0025-5718-96-00669-2
MathSciNet review: 1320898
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Abstract: Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli $k \le 72$ and other small moduli.


References [Enhancements On Off] (What's this?)

  • 1 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, with an appendix by P. Bateman, 3rd edition, Chelsea, New York, 1974. MR 16:904d.
  • 2 J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), 667--681. MR 87e:11102.
  • 3 K. S. McCurley, Explicit estimates for functions of primes in arithmetic progressions, Ph.D. thesis, University of Illinois at Urbana-Champagne, 1981.
  • 4 ------, Explicit zero-free regions for Dirichlet $L$-functions, J. Number Theory 19 (1984), 7--32.MR 85k:11041.
  • 5 ------, Explicit estimates for the error term in the prime number theorem for arithmetic progressions, Math. Comp. 42 (1984), 265--285. MR 85e:11065.
  • 6 ------, Explicit estimates for $\theta (X;3,l)$ and $\psi (X;3,l)$, Math. Comp. 42 (1984), 287--296. MR 85g:11085.
  • 7 W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical recipes, Cambridge Univ. Press, Cambridge, 1986. MR 87m:65001a.
  • 8 J. B. Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211--232. MR 2:150e.
  • 9 J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (X)$ and $\psi (X)$, Math. Comp. 29 (1975), 243--269. MR 56:15581a.
  • 10 R. Rumely, Numerical computations concerning the ERH, Math. Comp. 62 (1993), 415--440. MR 94b:11085.
  • 11 L. Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (X)$ and $\psi (X)$. II, Math. Comp. 30 (1976), 337--360. MR 56:15581b.
  • 12 S. B. Stechkin, Rational inequalities and zeros of the Riemann zeta-function, Trudy Mat. Inst. Steklov. 189 (1989), 110--116; English transl. in Proc. Steklov Inst. Math.. MR 90f:11071.
  • 13 R. Terras, A Miller algorithm for an incomplete Bessel function, J. Comput. Phys. 39 (1981), 233--240. MR 83e:65018.

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Additional Information

Olivier Ramaré
Affiliation: Département de Mathématiques, Université de Nancy I, URA 750, 54506 Van-doeuvre Cedex, France

Robert Rumely
Affiliation: addressDepartment of Mathematics, University of Georgia, Athens, Georgia 30602

DOI: https://doi.org/10.1090/S0025-5718-96-00669-2
Received by editor(s): February 26, 1993
Received by editor(s) in revised form: January 24, 1994, June 27, 1994, and January 10, 1995
Article copyright: © Copyright 1996 American Mathematical Society

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