<s>
In	O
mathematics	O
,	O
a	O
Redheffer	B-Algorithm
matrix	I-Algorithm
,	O
often	O
denoted	O
as	O
studied	O
by	O
,	O
is	O
a	O
square	B-Algorithm
(	O
0	O
,	O
1	O
)	O
matrix	O
whose	O
entries	O
aij	O
are	O
1	O
if	O
i	O
divides	O
j	O
or	O
if	O
j	O
=	O
1	O
;	O
otherwise	O
,	O
aij	O
=	O
0	O
.	O
</s>
<s>
It	O
is	O
useful	O
in	O
some	O
contexts	O
to	O
express	O
Dirichlet	O
convolution	O
,	O
or	O
convolved	O
divisors	O
sums	O
,	O
in	O
terms	O
of	O
matrix	O
products	O
involving	O
the	O
transpose	O
of	O
the	O
Redheffer	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
then	O
we	O
can	O
define	O
the	O
Redheffer	O
(	O
transpose	O
)	O
matrix	O
to	O
be	O
the	O
nxn	O
square	B-Algorithm
matrix	I-Algorithm
in	O
usual	O
matrix	O
notation	O
.	O
</s>
<s>
The	O
matrix	O
below	O
is	O
the	O
1212	O
Redheffer	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
determinant	O
of	O
the	O
nxn	O
square	B-Algorithm
Redheffer	B-Algorithm
matrix	I-Algorithm
is	O
given	O
by	O
the	O
Mertens	O
function	O
M(n )	O
.	O
</s>
<s>
This	O
results	O
in	O
an	O
interesting	O
characterization	O
that	O
the	O
Mertens	O
function	O
can	O
only	O
change	O
signs	O
infinitely	O
often	O
if	O
the	O
Redheffer	B-Algorithm
matrix	I-Algorithm
is	O
singular	O
at	O
infinitely	O
many	O
natural	O
numbers	O
,	O
which	O
is	O
widely	O
believed	O
to	O
be	O
the	O
case	O
with	O
respect	O
to	O
the	O
oscillatory	O
behavior	O
of	O
The	O
determinants	O
of	O
the	O
Redheffer	O
matrices	O
are	O
immediately	O
tied	O
to	O
the	O
Riemann	O
Hypothesis	O
(	O
RH	O
)	O
through	O
this	O
intimate	O
relation	O
with	O
the	O
Mertens	O
function	O
as	O
the	O
RH	O
is	O
equivalent	O
to	O
showing	O
that	O
for	O
all	O
(	O
sufficiently	O
small	O
)	O
.	O
</s>
<s>
This	O
observation	O
is	O
offered	O
in	O
so	O
much	O
as	O
for	O
a	O
fixed	O
arithmetic	B-Algorithm
function	I-Algorithm
f	O
,	O
the	O
coefficients	O
of	O
the	O
next	O
Lambert	O
series	O
expansion	O
over	O
f	O
provide	O
a	O
so-called	O
inclusion	O
mask	O
for	O
the	O
indices	O
over	O
which	O
we	O
sum	O
f	O
to	O
arrive	O
at	O
the	O
series	O
coefficients	O
of	O
these	O
expansions	O
.	O
</s>
<s>
In	O
particular	O
,	O
this	O
implies	O
that	O
is	O
not	O
diagonalizable	B-Algorithm
whenever	O
.	O
</s>
<s>
This	O
interpretation	O
is	O
a	O
great	O
deal	O
more	O
combinatorial	O
in	O
construction	O
than	O
typical	O
treatments	O
of	O
the	O
special	O
Redheffer	B-Algorithm
matrix	I-Algorithm
determinants	O
.	O
</s>
<s>
Before	O
diving	O
into	O
the	O
full	O
construction	O
of	O
this	O
spin	O
on	O
the	O
Redheffer	B-Algorithm
matrix	I-Algorithm
variants	O
defined	O
above	O
,	O
observe	O
that	O
this	O
type	O
of	O
expansion	O
is	O
in	O
many	O
ways	O
essentially	O
just	O
another	O
variation	O
of	O
the	O
usage	O
of	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
to	O
represent	O
truncated	O
power	O
series	O
expressions	O
where	O
the	O
matrix	O
entries	O
are	O
coefficients	O
of	O
the	O
formal	O
variable	O
in	O
the	O
series	O
.	O
</s>
<s>
See	O
the	O
citations	O
to	O
the	O
references	O
and	O
for	O
existing	O
generalizations	O
of	O
the	O
Redheffer	O
matrices	O
in	O
the	O
context	O
of	O
general	O
arithmetic	B-Algorithm
function	I-Algorithm
cases	O
.	O
</s>
<s>
First	O
,	O
given	O
any	O
two	O
non-identically-zero	O
arithmetic	B-Algorithm
functions	I-Algorithm
f	O
and	O
g	O
,	O
we	O
can	O
provide	O
explicit	O
matrix	O
representations	O
which	O
encode	O
their	O
Dirichlet	O
convolution	O
in	O
rows	O
indexed	O
by	O
natural	O
numbers	O
:	O
</s>
<s>
It	O
is	O
clear	O
that	O
in	O
general	O
the	O
Dirichlet	O
inverse	O
for	O
f	O
,	O
i.e.	O
,	O
the	O
uniquely	O
defined	O
arithmetic	B-Algorithm
function	I-Algorithm
such	O
that	O
,	O
involves	O
sums	O
of	O
nested	O
divisor	O
sums	O
of	O
depth	O
from	O
one	O
to	O
where	O
this	O
upper	O
bound	O
is	O
the	O
prime	O
omega	O
function	O
which	O
counts	O
the	O
number	O
of	O
distinct	O
prime	O
factors	O
of	O
n	O
.	O
As	O
this	O
example	O
shows	O
,	O
we	O
can	O
formulate	O
an	O
alternate	O
way	O
to	O
construct	O
the	O
Dirichlet	O
inverse	O
function	O
values	O
via	O
matrix	O
inversion	O
with	O
our	O
variant	O
Redheffer	O
matrices	O
,	O
.	O
</s>
<s>
Other	O
good	O
examples	O
of	O
this	O
type	O
of	O
factorization	O
treatment	O
to	O
inverting	O
relations	O
between	O
sums	O
over	O
sufficiently	O
invertible	O
,	O
or	O
well	O
enough	O
behaved	O
triangular	O
sets	O
of	O
weight	O
coefficients	O
include	O
the	O
Mobius	O
inversion	O
formula	O
,	O
the	O
binomial	O
transform	O
,	O
and	O
the	O
Stirling	B-Algorithm
transform	I-Algorithm
,	O
among	O
others	O
.	O
</s>
