Let
ϝ
(
n
)
\digamma (n)
denote a multiplicative function with range
{
−
1
,
0
,
1
}
\{-1,0,1\}
, and let
F
(
x
)
=
∑
n
=
1
⌊
x
⌋
ϝ
(
n
)
F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \digamma (n)
. Then
F
(
x
)
/
x
=
a
x
+
b
+
E
(
x
)
F(x)/\sqrt {x} = a\sqrt {x} + b + E(x)
, where
a
a
and
b
b
are constants and
E
(
x
)
E(x)
is an error term that either tends to
0
0
in the limit or is expected to oscillate about
0
0
in a roughly balanced manner. We say
F
(
x
)
F(x)
has persistent bias
b
b
(at the scale of
x
\sqrt {x}
) in the first case, and apparent bias
b
b
in the latter. For example, if
ϝ
(
n
)
=
μ
(
n
)
\digamma (n)=\mu (n)
, the Möbius function, then
F
(
x
)
=
∑
n
=
1
⌊
x
⌋
μ
(
n
)
F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \mu (n)
has apparent bias
0
0
, while if
ϝ
(
n
)
=
λ
(
n
)
\digamma (n)=\lambda (n)
, the Liouville function, then
F
(
x
)
=
∑
n
=
1
⌊
x
⌋
λ
(
n
)
F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \lambda (n)
has apparent bias
1
/
ζ
(
1
/
2
)
1/\zeta (1/2)
. We study the bias when
ϝ
(
p
k
)
\digamma (p^k)
is independent of the prime
p
p
, and call such functions fake
μ
′
s
\mu ’s
. We investigate the conditions required for such a function to exhibit a persistent or apparent bias, determine the functions in this family with maximal and minimal bias of each type, and characterize the functions with no bias. For such a function
F
(
x
)
F(x)
with apparent bias
b
b
, we also show that
F
(
x
)
/
x
−
a
x
−
b
F(x)/\sqrt {x}-a\sqrt {x}-b
changes sign infinitely often.