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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Explicit estimates for the error term in the prime number theorem for arithmetic progressions


Author: Kevin S. McCurley
Journal: Math. Comp. 42 (1984), 265-285
MSC: Primary 11N13; Secondary 11-04, 11Y35
MathSciNet review: 726004
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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 $ \vert\psi (x;k,l) - x/\varphi (k)\vert < \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 zero-free region for Dirichlet L-functions.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1984-0726004-6
PII: S 0025-5718(1984)0726004-6
Article copyright: © Copyright 1984 American Mathematical Society