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ARITHMETICAL FUNCTIONS I: MULTIPLICATIVE
FUNCTIONS
PETE L. CLARK
1.
Arithmetical Functions
Deﬁnition: An
arithmetical function
is a function
f
:
Z
+
→
C
.
Truth be told, this deﬁnition is a bit embarrassing. It would mean that taking any
function from calculus whose domain contains [1
,
+
∞
) and restricting it to positive
integer values, we get an arithmetical function. For instance,
e

3
x
cos
2
x
+(17 log(
x
+1))
is
an arithmetical function according to this deﬁnition, although it is, at best, dubious
whether this function holds any signiﬁcance in number theory.
If we were honest, the deﬁnition we would like to make is that an arithmetical
function is a real or complexvalued function deﬁned for positive integer arguments
which
is of some arithmetic signiﬁcance
, but of course this is not a formal deﬁnition
at all. Probably it is best to give examples:
Example
A
Ω: The prime counting function
n
7→
π
(
n
), the number of prime num
bers
p
, 1
≤
p
≤
n
.
This is the example
par excellence
of an arithmetical function: approximately half
of number theory is devoted to understanding its behavior. This function really
deserves a whole unit all to itself, and it will get one: we put it aside for now and
consider some other examples.
Example 1: The function
ω
(
n
), which counts the number of distinct prime di
visors of
n
.
Example 2: The function
ϕ
(
n
), which counts the number of integers
k
, 1
≤
k
≤
n
,
with gcd(
k,n
) = 1. Properly speaking this function is called
the totient func
tion
, but its fame inevitably precedes it and modern times it is usually called just
“the phi function” or “Euler’s phi function.” Since a congruence class
k
modulo
n
is invertible in the ring
Z
/n
Z
iﬀ its representative
k
is relatively prime to
n
, an
equivalent deﬁnition is
ϕ
(
n
) := #(
Z
/n
Z
)
×
,
the cardinality of the unit group of the ﬁnite ring
Z
/n
Z
.
Example 3: The function
n
7→
d
(
n
), the number of positive divisors of
n
.
1
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PETE L. CLARK
Example 4: For any integer
k
, the function
σ
k
(
n
), deﬁned as
σ
k
(
n
) =
X
d

n
d
k
,
the sum of the
k
th powers of the positive divisors of
n
. Note that
σ
0
(
n
) =
d
(
n
).
Example 5: The M¨obius function
μ
(
n
), deﬁned as follows:
μ
(1) = 1,
μ
(
n
) = 0
if
n
is not squarefree;
μ
(
p
1
···
p
r
) = (

1)
r
, when
p
1
,...,p
r
are distinct primes.
Example 6: For a positive integer
k
, the function
r
k
(
n
) which counts the num
ber of representations of
n
as a sum of
k
integral squares:
r
k
(
n
) = #
{
(
a
1
,...,a
k
)

a
2
1
+
...
+
a
2
k
=
n
}
.
These examples already suggest many others. Notably, all our examples but Ex
ample 5 are special cases of the following general construction: if we have on hand,
for any positive integer
n
, a ﬁnite set
S
n
of arithmetic objects, then we can deﬁne
an arithmetic function by deﬁning
n
7→
#
S
n
. This shows the link between number
theory and combinatorics. In fact the M¨obius function
μ
is a yet more purely com
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 Spring '11
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 Calculus, Number Theory

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