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Multiplicative Functions
: NumThy
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
squash@ufl.edu
Webpage
http://www.math.uﬂ.edu/
∼
squash/
23 March, 2011 (at
10:41
)
Aside.
In
number_theory.ams.tex
there is a
bit more on multiplicative functions also in
mult.
convolution.ams.tex
; run
collect.ams.tex
though.
See
generating_func.latex
for an application of the
M¨
obius fnc.
Basic
Given two arbitrary
♥
1
functions
f,g
:
Z
+
→
C
, de
ﬁne their
convolution
(
“Dirichlet convolution”
) by
h
f
~
g
i
(
K
) :=
X
a
·
b
=
K
f
(
a
)
·
g
(
b
)
.
Each such sum is to be interpreted as over all
ordered
pairs (
a,b
) of
positive
divisors of
K
. Eas
ily, convolution is commutative
♥
1
and associative.
Let
b
G
be the set of functions
f
:
Z
+
→
C
such
that
f
(1)
6
= 0. Inside is
G
⊂
b
G
, the set of
good
functions
, which have
f
(1) = 1. Say that a
good
f
is
multiplicative
if for all posints
♥
2
K
and
Γ
:
K
⊥
Γ
=
⇒
f
(
K
·
Γ
) =
f
(
K
)
·
f
(
Γ
)
.
♥
1
Indeed, they could map into a general ring. If this ring
is commutative then convolution will be commutative.
♥
2
Use
≡
N
to mean “congruent mod
N
”. Let
n
⊥
k
mean that
n
and
k
are coprime. Use
k
•
n
for “
k
divides
n
”. Its negation
k
±
r

n
means “
k
does not
divide
n
.” Use
n
•
k
and
n
r

±
k
for “
n
is/isnot a multiple of
k
.” Finally, for
p
a prime and
E
a natnum: Use doubleverticals,
p
E
•
n
,
to mean that
E
is the
highest
power of
p
which divides
n
.
Or write
n
•
p
E
to emphasize that this is an assertion
about
n
. Use
PoT
for
Power of Two
and
PoP
for
Power
of (
a
) Prime
.
For
N
a posint, let Φ(
N
) mean the set
{
r
∈
[
1
..N
]

r
⊥
N
}
. The cardinality
ϕ
(
N
) :=

Φ(
N
)

is the
Euler phi
function
. (So
ϕ
(
N
) is the cardinality of the multiplicative
group, Φ(
N
), in the
Z
N
ring.) Use
EFT
for the
Euler
Fermat Thm
, which says:
Suppose that integers
b
⊥
N
,
with
N
positive. Then
b
ϕ
(
N
)
≡
N
1
.
Let
M
⊂
G
be the set of multiplicative
♥
3
func
tions.
Basic functions.
Deﬁne fncs
δ,
1
,Id
∈
M
by:
δ
(1) := 1
and
δ
(
6
=1) := 0;
1
(
n
) := 1 and
Id
(
n
) :=
n
for all posints
n
.
Evidently
δ
() is a neutral element for convolution.
Also deﬁne, as
n
varies over the posints, these
MFs:
τ
(
n
) :=
X
d
:
d
•
n
1
and
σ
(
n
) :=
X
d
:
d
•
n
d.
This
τ
is called the (
number of
)
divisor
fnc, and
σ
is called the
sum of divisor
fnc. So
σ
(4) =
1 + 2 + 4 = 7 and
τ
(4) = 1 + 1 + 1 = 3.
Each of
δ,
1
,Id,
τ
,
σ
is multiplicative. So is Eu
ler
ϕ
; this will follow from (4).
1.0
:
Theorem.
(
b
G
,
~
,δ
)
is a commutative group
and
M
⊂
G
⊂
b
G
are subgroups.
♦
Proof.
(
The arguments showing
G
a group apply to show
b
G
a group. So we only argue for
G
and
M
.
)
Easily
G
is sealed under convolution. To show
the same for
M
, take fncs
f,h
∈
M
and posints
K
⊥
Γ
. Given posints (
x,y
) with
x
·
y
equaling
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