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

Abstract

We give explicit numerical estimates for the Chebyshev functions ψ ( x ; k , l ) \psi (x;k,l) and θ ( x ; k , l ) \theta (x;k,l) for certain nonexceptional moduli k. For values of ε \varepsilon and b, a constant c is tabulated such that | ψ ( x ; k , l ) − x / φ ( k ) | > ε x / φ ( k ) |\psi (x;k,l) - x/\varphi (k)| > \varepsilon x/\varphi (k) , provided ( k , l ) = 1 (k,l) = 1 , x ⩾ exp ⁡ ( c log 2 k ) x \geqslant \exp (c{\log ^2}k) , and k ⩾ 10 b k \geqslant {10^b} . The methods are similar to those used by Rosser and Schoenfeld in the case k = 1 k = 1 , but are based on explicit estimates of N ( T , χ ) N(T,\chi ) and an explicit zero-free region for Dirichlet L-functions.