Divisor function

Key takeaways

Divisor function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Divisor function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Divisor function from memory before moving on to harder problems.

Reference excerpt

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.

Definition The sum of positive divisors function σz(n), for a real or complex number z, is defined as the sum of the zth powers of the positive divisors of n. It can be expressed in sigma notation as

σ z ( n ) = ∑ d ∣ n d z , {\displaystyle \sigma _{z}(n)=\sum _{d\mid n}d^{z}\,\!,}

where d ∣ n {\displaystyle {d\mid n}} is shorthand for "d divides n". The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ0(n), or the number-of-divisors function (OEIS: A000005). When z is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so σ(n) is the same as σ1(n) (OEIS: A000203). The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, OEIS: A001065), and equals σ1(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.

Example For example, σ0(12) is the number of the divisors of 12:

σ 0 ( 12 ) = 1 0 + 2 0 + 3 0 + 4 0 + 6 0 + 12 0 = 1 + 1 + 1 + 1 + 1 + 1 = 6 , {\displaystyle {\begin{aligned}\sigma _{0}(12)&=1^{0}+2^{0}+3^{0}+4^{0}+6^{0}+12^{0}\\&=1+1+1+1+1+1=6,\end{aligned}}}

while σ1(12) is the sum of all the divisors:

σ 1 ( 12 ) = 1 1 + 2 1 + 3 1 + 4 1 + 6 1 + 12 1 = 1 + 2 + 3 + 4 + 6 + 12 = 28 , {\displaystyle {\begin{aligned}\sigma _{1}(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}+12^{1}\\&=1+2+3+4+6+12=28,\end{aligned}}}

and the aliquot sum s(12) of proper divisors is:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Divisor function

Start with the simplest possible case. Write down what Divisor function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Divisor function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Divisor function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Divisor function

In research: Divisor function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Divisor function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Divisor function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Analytic number theory, Divisor function, Number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Divisor function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Divisor function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Divisor function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Divisor function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Divisor function in simple terms?

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself).

Why does Divisor function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Divisor function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Divisor function.