Let
f
f
be a multiplicative arithmetic function satisfying
|
f
|
≤
1
\left | f \right | \leq 1
, let
x
≥
10
x \geq 10
and
2
≤
Q
≤
x
1
/
3
2 \leq Q \leq {x^{1/3}}
. It Is shown that, with suitable integers
q
1
≥
2
{q_1} \geq 2
and
q
2
≥
2
{q_2} \geq 2
, the estimate
\[
∑
n
≤
x
n
≡
a
mod
q
f
(
n
)
=
1
φ
(
q
)
∑
n
≤
x
(
n
,
q
)
=
1
f
(
n
)
+
O
(
x
q
(
log
log
x
log
Q
)
−
1
/
2
)
\sum \limits _{\begin {array}{*{20}{c}} {n \leq x} \\ {n \equiv a\bmod q} \\ \end {array} } {f(n) = \frac {1}{{\varphi (q)}}} \sum \limits _{\begin {array}{*{20}{c}} {n \leq x} \\ {(n,q) = 1} \\ \end {array} } {f(n) + O\left ( {\frac {x}{q}{{\left ( {\log \frac {{\log x}}{{\log Q}}} \right )}^{ - 1/2}}} \right )}
\]
holds uniformly for
(
a
,
q
)
=
1
\left ( {a,q} \right ) = 1
and all moduli
q
≤
Q
q \leq Q
that are not multiples of
q
1
{q_1}
or
q
2
{q_2}
.