Explicit Estimates for the Error Term in the Prime Number Theorem for Arithmetic Progressions

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 zero-free region for Dirichlet L-functions.