Explicit estimates for the error term in the prime number theorem for arithmetic progressions
HTML articles powered by AMS MathViewer
 by Kevin S. McCurley PDF
 Math. Comp. 42 (1984), 265285 Request permission
Abstract:
We give explicit numerical estimates for the Chebyshev functions $\psi (x;k,l)$ and $\theta (x;k,l)$ for certain nonexceptional moduli k. For values of $\varepsilon$ and b, a constant c is tabulated such that $\psi (x;k,l)  x/\varphi (k) < \varepsilon x/\varphi (k)$, provided $(k,l) = 1$, $x \geqslant \exp (c{\log ^2}k)$, and $k \geqslant {10^b}$. The methods are similar to those used by Rosser and Schoenfeld in the case $k = 1$, but are based on explicit estimates of $N(T,\chi )$ and an explicit zerofree region for Dirichlet Lfunctions.References

M. Abramowitz & I. Stegun, editors, Handbook of Mathematical Functions, Dover, New York, 1965.
R. Backlund, "Über die Nullstellen der Riemannschen Zetafunction," Acta Math., v. 41, 1918, pp. 374375.
 Harold Davenport, Multiplicative number theory, Lectures in Advanced Mathematics, No. 1, Markham Publishing Co., Chicago, Ill., 1967. Lectures given at the University of Michigan, Winter Term, 1966. MR 0217022
 A. E. Ingham, The distribution of prime numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR 1074573
 Kevin S. McCurley, Explicit zerofree regions for Dirichlet $L$functions, J. Number Theory 19 (1984), no. 1, 7–32. MR 751161, DOI 10.1016/0022314X(84)900891
 F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1974. MR 0435697
 Karl Prachar, Primzahlverteilung, SpringerVerlag, BerlinGöttingenHeidelberg, 1957 (German). MR 0087685
 Hans Rademacher, On the PhragménLindelöf theorem and some applications, Math. Z 72 (1959/1960), 192–204. MR 0117200, DOI 10.1007/BF01162949
 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, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$, Math. Comp. 29 (1975), 243–269. MR 457373, DOI 10.1090/S00255718197504573737
 J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$, Math. Comp. 29 (1975), 243–269. MR 457373, DOI 10.1090/S00255718197504573737
 Riho Terras, A Miller algorithm for an incomplete Bessel function, J. Comput. Phys. 39 (1981), no. 1, 233–240. MR 608723, DOI 10.1016/00219991(81)901479
Additional Information
 © Copyright 1984 American Mathematical Society
 Journal: Math. Comp. 42 (1984), 265285
 MSC: Primary 11N13; Secondary 1104, 11Y35
 DOI: https://doi.org/10.1090/S00255718198407260046
 MathSciNet review: 726004