Landau's function

Key takeaways

Landau's 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 Landau's function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Landau's function from memory before moving on to harder problems.

Reference excerpt

In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn. Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation of n elements can be recursively applied to itself before it returns to its starting sequence. For instance, 5 = 2 + 3 and lcm(2,3) = 6. No other partition of 5 yields a bigger lcm, so g(5) = 6. An element of order 6 in the group S5 can be written in cycle notation as (1 2) (3 4 5). Note that the same argument applies to the number 6, that is, g(6) = 6. There are arbitrarily long sequences of consecutive numbers n, n + 1, ..., n + m on which the function g is constant. The integer sequence g(0) = 1, g(1) = 1, g(2) = 2, g(3) = 3, g(4) = 4, g(5) = 6, g(6) = 6, g(7) = 12, g(8) = 15, ... (sequence A000793 in the OEIS) is named after Edmund Landau, who proved in 1902 that

lim n → ∞ ln ⁡ ( g ( n ) ) n ln ⁡ ( n ) = 1 {\displaystyle \lim _{n\to \infty }{\frac {\ln(g(n))}{\sqrt {n\ln(n)}}}=1}

(where ln denotes the natural logarithm). Equivalently (using little-o notation), g ( n ) = e ( 1 + o ( 1 ) ) n ln ⁡ n {\displaystyle g(n)=e^{(1+o(1)){\sqrt {n\ln n}}}} . More precisely,

ln ⁡ g ( n ) = n ln ⁡ n ( 1 + ln ⁡ ln ⁡ n − 1 2 ln ⁡ n − ( ln ⁡ ln ⁡ n ) 2 − 6 ln ⁡ ln ⁡ n + 9 8 ( ln ⁡ n ) 2 + O ( ( ln ⁡ ln ⁡ n ln ⁡ n ) 3 ) ) . {\displaystyle \ln g(n)={\sqrt {n\ln n}}\left(1+{\frac {\ln \ln n-1}{2\ln n}}-{\frac {(\ln \ln n)^{2}-6\ln \ln n+9}{8(\ln n)^{2}}}+O\left(\left({\frac {\ln \ln n}{\ln n}}\right)^{3}\right)\right).}

If π ( x ) − Li ⁡ ( x ) = O ( R ( x ) ) {\displaystyle \pi (x)-\operatorname {Li} (x)=O(R(x))} , where π {\displaystyle \pi } denotes the prime counting function, Li {\displaystyle \operatorname {Li} } the logarithmic integral function with inverse Li − 1 {\displaystyle \operatorname {Li} ^{-1}} , and we may take R ( x ) = x exp ⁡ ( − c ( ln ⁡ x ) 3 / 5 ( ln ⁡ ln ⁡ x ) − 1 / 5 ) {\displaystyle R(x)=x\exp {\bigl (}-c(\ln x)^{3/5}(\ln \ln x)^{-1/5}{\bigr )}} for some constant c > 0 by Ford, then

ln ⁡ g ( n ) = Li − 1 ⁡ ( n ) + O ( R ( n ln ⁡ n ) ln ⁡ n ) . {\displaystyle \ln g(n)={\sqrt {\operatorname {Li} ^{-1}(n)}}+O{\bigl (}R({\sqrt {n\ln n}})\ln n{\bigr )}.}

The statement that

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Landau's function

Start with the simplest possible case. Write down what Landau's 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 Landau's 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 Landau's 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 Landau's function

In research: Landau's 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 Landau's 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: Landau's function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arithmetic functions, Group theory, Permutations, so understanding it makes those chapters shorter.
In everyday life: Look for Landau's 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 Landau's function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Landau's 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 Landau's function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Landau's function in simple terms?

In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn. Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation…

Why does Landau's 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 Landau's 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 Landau's function.