For an additive arithmetic function f, and positive integer D, let E(x,D) be \[ \max_{y\leq x}\max_{(r,D)=1}| \sum_{n\leq y,\quad n\equiv r (mod D)}f(n)-(1/\phi (D))\sum_{n\leq y,\quad (n,D)=1}f(n)|. \] Strengthening results from Chapter 7 of his monograph ''Arithmetic functions and integer products'' (1985; Zbl 0559.10032), the author proves that for every fixed \(\alpha <1/2\) \(\sum_{q\leq x^{\alpha}}\phi (q)(E(x,q))^ 2\ll x^ 2((\log \log x)^ 4/\log x)\sum_{q\leq x}| f(q)|^ 2/q,\) where the summations run over prime-power moduli q. It is important that this inequality is uniform in f; it is abstract in form. If required, extra conditions \(n\equiv r (mod c)\) may be built into the definition of E(x,D), provided the \(q\leq x^{\alpha}\) are restricted to those coprime to c. Apart from the term \((\log\log x)^ 4\), the factor in the upper bound is best-possible. Perhaps \(\alpha <1\) is true. Whilst this result is like the Bombieri-Vinogradov theorem [\textit{E. Bombieri}, Mathematika 12, 201-225 (1965; Zbl 0136.330)] in appearance, unlike the standard methods for that theorem, the proof does not proceed by way of \(L^ 1\)-estimates. \(L^ 2\)-arguments are used from the outset. In particular, a Hilbert space version of the large sieve inequality with wide uniformity is employed, together with a Fourier inversion by means of Dirichlet series.