In this chapter, we shall set up the basic theory of arithmetic functions. This theory will be seen in action in later chapters, but in particular in chapter 9.
Definition 2.1:
An arithmetical function is a function
.
Definition 2.2 (important arithmetical functions):
- The Kronecker delta:

- Euler's totient function:

- Möbius'
-function: 
- The von Mangoldt function:

- The monomials:

- The number of distinct prime divisors:
, 
- The sum of prime factors with multiplicity:
, 
- The Liouville function:

- Exercise 2.1.1: Compute
,
and
.
- Exercise 2.1.2: Compute
. Hint:
.
- Exercise 2.1.3: Compute
up to three decimal places. Hint: Use a Taylor expansion.
- Exercise 2.1.4: Prove that for each
and
.
In the following theorem, we show that the arithmetical functions form an Abelian monoid, where the monoid operation is given by the convolution. Further, since the sum of two arithmetic functions is again an arithmetic function, the arithmetic functions form a commutative ring. In fact, as we shall also see, they form an integral domain.
Proof:
1.:
,
where
is a bijection from the set of divisors of
to itself.
2.:
,
where the last equality follows from the identity function

being a bijection. But

and hence associativity.
3.:


Theorem 2.5:
The ring of arithmetic functions is an integral domain.
Proof:
Let
be arithmetic functions, and let
be minimal such that
,
. Then
.
We shall now determine the units of the ring of arithmetic functions.
Theorem 2.6:
Let
be an arithmetic function. Then
is invertible (with respect to convolution) if and only if
.
Proof:
Assume first
. Then for any arithmetic function
,
.
Assume now
. Then
, given by the recursive formula
,
, 
is an inverse (and thus the inverse) of
, since
and for
inductively

- Exercise 2.2.1:
- Exercise 2.2.2:
Definition 2.7:
An arithmetical function
is called multiplicative iff it satisfies
, and
.
Theorem 2.8:
Let
be multiplicative arithmetical functions. Then
is multiplicative.
Proof:
Let
. Then
,
since the function
is a bijection from the divisors of
to the Cartesian product of the divisors of
and the divisors of
; this is because multiplication is the inverse:
,
.
To rigorously prove this actually is an exercise in itself. But due to the multiplicativity of
and
,
.
Furthermore,
.
Since
is multiplicative, we conclude that the multiplicative functions form an Abelian submonoid of the arithmetic functions with convolution. Unfortunately, we do not have a subring since the sum of two multiplicative functions is never multiplicative (look at
).
Theorem 2.9:
Let
be a multiplicative function such that
converges absolutely. Then
.
Proof: Let
be the ordered sequence of all prime numbers. For all
we have

due to the multiplicativity of
. For each
, we successively take
, ...,
and then
. It follows from the definitions and the rule
that the right hand side converges to
.
We claim that
.
Indeed, choose
such that
.
Then by the fundamental theorem of arithmetic, there exists an
and
such that
.
Then we have by the triangle inequality for
,
and
arbitrary that

From this easily follows the claim.
It is left to show that the product on the left is independent of the order of multiplication. But this is clear since if the sequence
is enumerated differently, the argument works in just the same way and the left hand side remains the same.
Definition 2.10:
An arithmetical function
is called strongly multiplicative iff it satisfies
, and
.
Equivalently, a strongly multiplicative function is a monoid homomorphism
.
Theorem 2.11:
Let
be a strongly multiplicative function such that
converges absolutely. Then
.
Proof:
Due to theorem 2.9, we have
.
Due to strong multiplicativity and the geometric series, the latter expression equals
.
- Exercise 2.3.1: Let
be an arithmetic function such that for all
, and let
. Prove that the function
is multiplicative.
Examples 2.13:
We shall here compute the Bell series for some important arithmetic functions.
We note that in general for a completely multiplicative function
, we have
.
In particular, in this case the Bell series defines a function.
1. The Kronecker delta:

2. Euler' totient function (we use lemma 9.?):

3. The Möbius
function:

4. The von Mangoldt function:

5. The monomials:

6. The number of distinct prime divisors:

7. The number of prime divisors including multiplicity:

8. The Liouville function:

Theorem 2.14 (compatibility of Bell series and convolution):
Let
arithmetic functions, and
be a prime. Then
.
Proof:


In case of multiplicativity, we have the following theorem:
Theorem 2.15 (Uniqueness theorem):
Let
be multiplicative functions. Then
.
Proof:
is pretty obvious;
:
as formal power series is equivalent to saying
. If now
, then

due to the multiplicativity of
and
.
In chapter 9, we will use Bell series to obtain equations for number-theoretic functions.
Definition 2.16:
Let
be an arithmetic function. Then the derivative of
is defined to be the function
.
Proof:
1. is easily checked.
2.:

3.
We have
and
. Hence, by 2.
.
Convolving with
and using
yields the desired formula.
Note that a chain rule wouldn't make much sense, since
arithmetic may map anywhere but to
and thus
doesn't make a lot of sense in general.