×

Primes in arithmetic progressions. (English) Zbl 0856.11042

This paper presents explicit error bounds for certain cases of the prime number theorem for arithmetic progressions. Define \[ \theta(x;k,l)= \sum_{p\equiv l\bmod k,\;p\leq x}\log p, \] where the sum is over primes \(p\), and \[ \psi(x;k,l)= \sum_{n\equiv l\bmod k,\;n\leq x}\Lambda(n), \] where \(\Lambda\) denotes Von Mangoldt’s function. The authors give various explicit error estimates for \(\theta\) and \(\psi\) for \(k\leq 72\), or \(k\leq 112\) and composite, or \(k=116\), 117, 120, 121, 124, 125, 128, 132, 140, 143, 144, 156, 163, 169, 180, 216, 243, 256, 360, 420, 432. The proofs of these bounds utilize method, of McCurley, Rosser, and Schoenfeld using zero-free regions for certain Dirichlet \(L\)-functions as computed by the second author.

MSC:

11N13 Primes in congruence classes
11Y35 Analytic computations
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11-04 Software, source code, etc. for problems pertaining to number theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, with an appendix by P. Bateman, 3rd edition, Chelsea, New York, 1974. . · Zbl 0051.28007
[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), no. 174, 667 – 681. · Zbl 0585.10023
[3] K. S. McCurley, Explicit estimates for functions of primes in arithmetic progressions, Ph.D. thesis, University of Illinois at Urbana-Champagne, 1981.
[4] Kevin S. McCurley, Explicit zero-free regions for Dirichlet \?-functions, J. Number Theory 19 (1984), no. 1, 7 – 32. · Zbl 0536.10035 · doi:10.1016/0022-314X(84)90089-1
[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. · Zbl 0535.10043
[6] Kevin S. McCurley, Explicit estimates for \?(\?;3,\?) and \?(\?;3,\?), Math. Comp. 42 (1984), no. 165, 287 – 296. · Zbl 0535.10044
[7] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling, Numerical recipes, Cambridge University Press, Cambridge, 1986. The art of scientific computing. William T. Vetterling, Saul A. Teukolsky, William H. Press, and Brian P. Flannery, Numerical recipes example book (Pascal), Cambridge University Press, Cambridge, 1985. William T. Vetterling, Saul A. Teukolsky, William H. Press, and Brian P. Flannery, Numerical recipes example book (FORTRAN), Cambridge University Press, Cambridge, 1985. · Zbl 0587.65003
[8] J. B. Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211–232. . · JFM 67.0129.03
[9] J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions \?(\?) and \?(\?), Math. Comp. 29 (1975), 243 – 269. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. , https://doi.org/10.1090/S0025-5718-1975-0457373-7 Lowell Schoenfeld, Sharper bounds for the Chebyshev functions \?(\?) and \?(\?). II, Math. Comp. 30 (1976), no. 134, 337 – 360. , https://doi.org/10.1090/S0025-5718-1976-0457374-X Lowell Schoenfeld, Corrigendum: ”Sharper bounds for the Chebyshev functions \?(\?) and \?(\?). II” (Math. Comput. 30 (1976), no. 134, 337 – 360), Math. Comp. 30 (1976), no. 136, 900.
[10] Robert Rumely, Numerical computations concerning the ERH, Math. Comp. 61 (1993), no. 203, 415 – 440, S17 – S23. · Zbl 0792.11034
[11] J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions \?(\?) and \?(\?), Math. Comp. 29 (1975), 243 – 269. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. , https://doi.org/10.1090/S0025-5718-1975-0457373-7 Lowell Schoenfeld, Sharper bounds for the Chebyshev functions \?(\?) and \?(\?). II, Math. Comp. 30 (1976), no. 134, 337 – 360. , https://doi.org/10.1090/S0025-5718-1976-0457374-X Lowell Schoenfeld, Corrigendum: ”Sharper bounds for the Chebyshev functions \?(\?) and \?(\?). II” (Math. Comput. 30 (1976), no. 134, 337 – 360), Math. Comp. 30 (1976), no. 136, 900.
[12] S. B. Stechkin, Rational inequalities and zeros of the Riemann zeta function, Trudy Mat. Inst. Steklov. 189 (1989), 110 – 116 (Russian). Translated in Proc. Steklov Inst. Math. 1990, no. 4, 127 – 134; A collection of papers from the All-Union School on the Theory of Functions (Russian) (Dushanbe, 1986).
[13] G. W. Stewart, The efficient generation of random orthogonal matrices with an application to condition estimators, SIAM J. Numer. Anal. 17 (1980), no. 3, 403 – 409 (loose microfiche suppl.). · Zbl 0443.65027 · doi:10.1137/0717034
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.