Our best unconditional bounds for ψ(x, χ), such as Theorem 11.16, are not very good owing to our rather limited knowledge of the zero-free region of L(s, χ). It is possible to extend the range for the modulus q of χ at the expense of the error term or by including zeros close to 1. See Corollary 11.20 or the chapter on Vinogradov’s mean value theorem in the forthcoming Volume III. If we assume GRH, then we have a much better estimate (cf. Theorem 13.7). However, in some situations, a good bound for an average of |ψ (x, χ) | is all that is required, and such bounds can be obtained unconditionally by combining our methods of Chapter 17 with the large sieve.