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

Abstract

We give explicit numerical estimates for the Chebyshev functions <¡/(x; k.l) and 9{x; k, I) for certain nonexceptional moduli k.For values of e andb, a constant c is tabulated such that \ip(x; k, I) -x/<p(fc)| < ex/tp(k), provided (&,/) = 1, x > exp(clog2 k), and k > 10*.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,\) and an explicit zero-free region for Dirichlet /.-functions.

Locations

• Mathematics of Computation - View - PDF