Simple Arithmetic Functions

Recall from earlier classes that a function is a rule that assigns to every element of one set (the domain) exactly one element of another set (the range) — a kind of machine that converts an input, or pre-image, into a corresponding output, or image. For example, $y = x^2$ is a function in which $x$ is the pre-image and $y$ is the image.

A simple arithmetic function (also called a number-theoretic function) is a special case of this idea: its domain is restricted to the positive integers, while its range consists of real or complex numbers. Such functions are written as $f : \mathbb{Z}^+ \to \mathbb{R}$, and they exist to describe arithmetic properties of numbers — patterns like divisibility, primality, or coprimality — which makes them a core tool of number theory.

One widely used example is Euler's totient function, also called Euler's phi function and denoted $\phi(n)$. For a positive integer $n$, $\phi(n)$ counts how many integers in the set $\{1, 2, 3, \ldots, n\}$ are coprime to $n$ — that is, numbers whose greatest common divisor (GCD) with $n$ equals $1$.

Because it measures how many numbers up to $n$ share no common factor with $n$, the totient function is a useful gauge of how "prime-like" a number's neighbourhood is, and it recurs throughout number theory and its applications.

Additive and multiplicative functions

Arithmetic functions are often classified by how they behave on a product of two coprime numbers:

• A function $f$ is additive if $f(xy) = f(x) + f(y)$ whenever $\gcd(x, y) = 1$. The logarithm is a familiar model, since $\log(xy) = \log x + \log y$.
• A function $f$ is multiplicative if $f(xy) = f(x)\times f(y)$ whenever $\gcd(x, y) = 1$. Euler's totient is multiplicative: $\varphi(6) = \varphi(2)\times\varphi(3) = 1 \times 2 = 2$.

Properties of Euler's totient function

Three properties make $\varphi(n)$ easy to compute:

For example, $12 = 2^{2} \times 3$, so $\varphi(12) = 12\left(1 - \tfrac12\right)\left(1 - \tfrac13\right) = 12 \times \tfrac12 \times \tfrac23 = 4$.

The number-of-divisors function $\tau(n)$

Denoted by the Greek letter $\tau$ ("tau"), this function counts how many positive divisors $n$ has:

For instance $\tau(1) = 1$, $\tau(3) = 2$ (divisors $1, 3$) and $\tau(12) = 6$ (divisors $1, 2, 3, 4, 6, 12$).

The divisor-sum function $\sigma(n)$

Written with the Greek letter $\sigma$ ("sigma"), this function adds up all the positive divisors of $n$, including $1$ and $n$ itself:

So $\sigma(3) = 1 + 3 = 4$ and $\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$. For any prime $p$ the two functions take simple values, $\tau(p) = 2$ and $\sigma(p) = p + 1$, because a prime has exactly the two divisors $1$ and $p$.

The Mobius function $\mu(n)$

The Mobius function, written with the Greek letter $\mu$ ("mu"), records the prime-factor structure of $n$ using just three values:

For example, $12 = 2^{2} \times 3$ has the repeated prime factor $2$, so $\mu(12) = 0$; while $35 = 5 \times 7$ is a product of two distinct primes, giving $\mu(35) = (-1)^{2} = 1$. Both $\tau(n)$ and $\sigma(n)$ are multiplicative functions, which lets each be evaluated one prime power at a time.

Illustration 8. For $n = 1$, the only positive divisor is $1$ itself, so

Hence $\tau(n) = \sigma(n)$ when $n = 1$ — the count of divisors and the sum of divisors coincide, because there is a single divisor and it equals $1$.

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Illustration 9. Complete the table for prime numbers $n$, where $\tau(n)$ is the number of positive divisors and $\sigma(n)$ is their sum.

A prime $n$ has exactly two positive divisors, $1$ and $n$ itself. Therefore, for every prime:

$\tau(n) = 2 \qquad \text{and} \qquad \sigma(n) = 1 + n = n + 1.$ …