Multiplicative Functions

Multiplicative functions exhibit remarkable structural properties that make them central objects in analytic number theory. Their behavior on prime powers determines them completely.

DefinitionMultiplicative and Completely Multiplicative

A function $f: \mathbb{N} \to \mathbb{C}$ is multiplicative if $f(1) = 1$ and $f(mn) = f(m)f(n)$ whenever $\gcd(m,n) = 1$ .

It is completely multiplicative if $f(mn) = f(m)f(n)$ for all positive integers $m, n$ (not just coprime ones).

ExampleKey Multiplicative Functions

Euler's totient: $\varphi(p^k) = p^{k-1}(p-1)$ for prime $p$
Divisor sum: $\sigma(n) = \sum_{d|n} d$ , with $\sigma(p^k) = \frac{p^{k+1}-1}{p-1}$
Number of divisors: $\tau(n) = \sum_{d|n} 1$ , with $\tau(p^k) = k+1$
Möbius function: $\mu(p) = -1$ for all primes $p$ , $\mu(p^k) = 0$ for $k \geq 2$

The key insight is that multiplicative functions are completely determined by their values on prime powers. If $n = p_1^{a_1} p_2^{a_2} \cdots p_r^{a_r}$ is the prime factorization, then:

f(n) = f(p_1^{a_1}) f(p_2^{a_2}) \cdots f(p_r^{a_r})

DefinitionConvolution of Multiplicative Functions

If $f$ and $g$ are multiplicative, then their Dirichlet convolution $h = f * g$ is also multiplicative. At prime powers:

h(p^k) = \sum_{j=0}^{k} f(p^j) g(p^{k-j})

Remark

The convolution formula for multiplicative functions reduces computation from summing over all divisors to summing over prime power divisors. This is computationally advantageous and theoretically illuminating.

ExampleEuler's Totient via Convolution

The totient function satisfies $\varphi * 1 = \text{id}$ , meaning:

\sum_{d|n} \varphi(d) = n

This can be verified by counting: each integer $1 \leq k \leq n$ contributes $1$ to $\varphi(\gcd(k,n))$ .

Understanding multiplicativity provides both computational tools (reducing calculations to prime powers) and theoretical insights (algebraic structure under convolution). Many deep results in number theory rely on analyzing multiplicative functions through their Dirichlet series and exploiting their algebraic properties.