[Introduction to Number Theory Series Part 18] How Do Arithmetic Functions Summarize the Properties of Integers?

So far we have often focused on one integer or on a pair of integers. Arithmetic functions give us another viewpoint: they assign a numerical summary to each positive integer.

What this post covers

• What arithmetic functions are
• Three representative examples: \varphi(n), \tau(n), and \sigma(n)
• Why prime factorization sits behind them
• Why the function viewpoint is useful

Key terms

• arithmetic function: a function on the positive integers that records number-theoretic information
• Euler's totient function: the number of integers coprime to n
• divisor: a number that divides an integer exactly
• prime factorization: expressing an integer as a product of primes

What is an arithmetic function?

An arithmetic function takes a positive integer n and returns a value that summarizes some property of n.

Examples include:

• \varphi(n): number of positive integers up to n that are coprime to n
• \tau(n): number of positive divisors of n
• \sigma(n): sum of positive divisors of n

These are easier to understand if we ask first: what is this function counting or summing?

Example 1: the divisor-count function

Since

$$12=2^2\times 3,$$

we get

$$\tau(12)=(2+1)(1+1)=6.$$

Indeed, the divisors are

$$1,2,3,4,6,12.$$

Example 2: the divisor-sum function

For 12,

$$\sigma(12)=1+2+3+4+6+12=28.$$

Example 3: Euler's totient function

As we saw earlier,

$$\varphi(10)=4,$$

because 1, 3, 7, and 9 are coprime to 10.

Why is prime factorization so important here?

These functions may look different on the surface, but their actual computation is often controlled by prime factorization.

Once an integer is broken into prime powers, we can organize:

• how many divisors it has
• how those divisors add up
• how many numbers are coprime to it

So prime factorization is the common structural base.

Why is the function viewpoint useful?

Earlier posts often asked: “What is true for this one number?” Arithmetic functions make it easier to ask: “How do many integers compare?”

Some integers have many divisors. Some have very few. Some have many coprime neighbors relative to their size. A function packages that information into a comparable value.

Common mistakes

1. Memorizing symbols without the meaning

Always ask what the function counts or sums.

2. Treating arithmetic functions as a completely separate topic

They are tightly connected to divisors, coprimality, and prime factorization.

3. Trying to count everything directly every time

For larger numbers, structural formulas matter much more than manual listing.

Wrap-up

Arithmetic functions summarize the structure of integers numerically. They give a compact way to compare many integers through divisor counts, divisor sums, and coprime counts.

Next we take a brief look at a deeper topic: quadratic congruences and quadratic residues.