A complex valued function
$g$
, defined on the positive integers, is multiplicative if it satisfies
$g(ab) = g(a)g(b)$
whenever the integers
$a$
and
$b$
are mutually prime.
THEOREM 1. Let
$D$
be an integer,
$2 \le D \le x, \varepsilon > 0$
. Let
$g$
be a multiplicative function with values in the complex unit disc.
There is a character
$\chi_1({\rm mod}\, D)$
, real if
$g$
is real, such that when
$0 < \gamma < 1$
,
\[
\sum_{\overset{n \le y}{n\equiv a({\rm mod}\, D)}} g(n)- \frac{1}{\phi(D)}
\sum_{\overset{n\le y}{(n,D)=1)}} g(n)-\frac{\overline{\chi_1(a)}}{\phi(D)} \sum_{n\le y} g(n)\chi_1(n) \ll \frac{y}{\phi(D)} \left(\frac{\log D}{\log y}\right)^{1/4-\varepsilon} \]
uniformly for
$(a, D) = 1, D \le y, x^\gamma \le y \le x$
, the implied constant depending at most upon
$\varepsilon, \gamma$
.