Completely Multiplicative Functions

Multiplicative and Completely Multiplicative Functions

An arithmetic function is a function defined on the positive integers. Such a function

f:\mathbb{N}\to\mathbb{C}

is called multiplicative if

f(ab)=f(a)f(b)

whenever

\gcd(a,b)=1.

It is called completely multiplicative if the same identity holds for all positive integers $a$ and $b$ , without assuming coprimality:

f(ab)=f(a)f(b).

Thus complete multiplicativity is stronger than multiplicativity.

Basic Examples

The constant function

f(n)=1

is completely multiplicative.

The identity function

f(n)=n

is also completely multiplicative, since

f(ab)=ab=f(a)f(b).

For a fixed real or complex number $s$ , the function

f(n)=n^s

is completely multiplicative.

The Liouville function is another important example:

\lambda(n)=(-1)^{\Omega(n)}.

Since

\Omega(ab)=\Omega(a)+\Omega(b),

we have

\lambda(ab)=\lambda(a)\lambda(b)

for all positive integers $a,b$ .

Determination by Prime Values

A completely multiplicative function is determined entirely by its values on primes.

n=p_1^{\alpha_1}\cdots p_r^{\alpha_r},

then complete multiplicativity gives

f(n)=f(p_1)^{\alpha_1}\cdots f(p_r)^{\alpha_r}.

Thus once the values

f(p)

are known for all primes $p$ , the function is known on every positive integer.

For example, if

f(p)=-1

for every prime $p$ , then

f(n)=(-1)^{\Omega(n)}=\lambda(n).

Difference from Ordinary Multiplicativity

For an ordinary multiplicative function, one only has

f(ab)=f(a)f(b)

when $a$ and $b$ are coprime.

Thus the values on prime powers must be specified separately:

f(p), f(p^2), f(p^3),\ldots

For a completely multiplicative function, however,

f(p^\alpha)=f(p)^\alpha.

This is a strong restriction.

For example, the Möbius function is multiplicative but not completely multiplicative. We have

\mu(2)=-1,

so complete multiplicativity would imply

\mu(4)=\mu(2)^2=1.

But actually

\mu(4)=0.

Therefore $\mu$ is not completely multiplicative.

Dirichlet Series

Completely multiplicative functions have especially simple Dirichlet series.

Suppose $f$ is completely multiplicative and the series converges absolutely. Then

\sum_{n=1}^{\infty}\frac{f(n)}{n^s} = \prod_p \left(1+\frac{f(p)}{p^s}+\frac{f(p)^2}{p^{2s}}+\cdots\right).

Since the local series is geometric,

1+\frac{f(p)}{p^s}+\frac{f(p)^2}{p^{2s}}+\cdots = \frac{1}{1-f(p)p^{-s}}.

Therefore

\sum_{n=1}^{\infty}\frac{f(n)}{n^s} = \prod_p \frac{1}{1-f(p)p^{-s}}.

This Euler product is simpler than the general multiplicative case because each local factor is determined by a single value $f(p)$ .

Characters as Examples

Dirichlet characters are important examples of completely multiplicative functions, after being extended periodically and with zeros on non-coprime integers.

A Dirichlet character modulo $n$ is a function $\chi$ satisfying

\chi(ab)=\chi(a)\chi(b)

for all integers $a,b$ , together with periodicity modulo $n$ .

Such functions are central in the study of primes in arithmetic progressions and Dirichlet $L$ -functions.

They show that complete multiplicativity is not merely a formal property. It encodes arithmetic symmetries.

Cancellation

Many completely multiplicative functions take values on the unit circle or among signs.

For such functions, sums like

\sum_{n\le x} f(n)

measure cancellation. If $f(n)$ behaves randomly, positive and negative or complex values should partly cancel.

For the Liouville function, this sum is

\sum_{n\le x}\lambda(n).

Understanding its cancellation is connected to deep questions about prime factorization and the zeros of the zeta function.

Role in Number Theory

Completely multiplicative functions are basic objects in multiplicative number theory. They translate multiplication of integers directly into multiplication of function values.

Their structure is rigid: values on primes determine everything. This makes them algebraically simple and analytically useful.

They appear in Euler products, Dirichlet characters, $L$ -functions, sign patterns of prime factors, and cancellation problems. Complete multiplicativity is therefore one of the cleanest ways to encode prime factorization into arithmetic functions.