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Ramanujan tau function

Ramanujan tau function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ramanujan tau function rather than just read about it. In short: In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z} } defined by ∑ n = 1 ∞ τ ( n ) q n = q ∏ n = 1 ∞ ( 1 − q n ) 24 = q ϕ ( q ) 24 = η ( z ) 24 = Δ ( z ) , {\displaystyle \sum _{n=1}^{\infty }\tau (n)q^{n}=q\prod _{n=1}^{\infty }(1-q^{n})^{24}=q\phi (q)^{24}=\eta (z)^{24}=\Delta (z),} where ϕ {\displaystyle \phi } is t…

Ramanujan tau function — main illustration
Ramanujan tau function — illustration

Key takeaways

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

Reference excerpt

In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z} } defined by

∑ n = 1 ∞ τ ( n ) q n = q ∏ n = 1 ∞ ( 1 − q n ) 24 = q ϕ ( q ) 24 = η ( z ) 24 = Δ ( z ) , {\displaystyle \sum _{n=1}^{\infty }\tau (n)q^{n}=q\prod _{n=1}^{\infty }(1-q^{n})^{24}=q\phi (q)^{24}=\eta (z)^{24}=\Delta (z),}

where ϕ {\displaystyle \phi } is the Euler function, η {\displaystyle \eta } is the Dedekind eta function, Δ ( z ) {\displaystyle \Delta (z)} is the modular discriminant, and q = e 2 π i z {\displaystyle q=e^{2\pi iz}} with I m ( z ) > 0 {\displaystyle \mathrm {Im} (z)>0} .

Values The first few values of the tau function are given in the following table (sequence A000594 in the OEIS):

Calculating this function on an odd square number yields an odd number, whereas for any other number the function yields an even number.

Main properties Ramanujan conjectured two properties of τ ( n ) {\displaystyle \tau (n)} , which can be rephrased equivalently as:

τ ⁡ ( m n ) = τ ⁡ ( m ) τ ⁡ ( n ) {\displaystyle \mathop {\tau } (mn)=\mathop {\tau } (m)\mathop {\tau } (n)} if m {\displaystyle m} and n {\displaystyle n} are coprime (that is, τ ( n ) {\displaystyle \tau (n)} is a multiplicative function)

τ ( p r + 1 ) = τ ⁡ ( p ) τ ⁡ ( p r ) − p 11 τ ⁡ ( p r − 1 ) {\displaystyle \tau (p^{r+1})=\mathop {\tau } (p)\mathop {\tau } (p^{r})-p^{11}\mathop {\tau } (p^{r-1})} for p {\displaystyle p} prime and r > 0 {\displaystyle r>0} . Together, these two properties are equivalent to the identity

τ ⁡ ( m ) τ ⁡ ( n ) = ∑ d | ( m , n ) d 11 τ ⁡ ( m n d 2 ) , m , n ≥ 1. {\displaystyle \mathop {\tau } (m)\mathop {\tau } (n)=\sum _{d|(m,n)}d^{11}\mathop {\tau } \left({\frac {mn}{d^{2}}}\right),\quad m,n\geq 1.}

They were proved by Louis Mordell using what is now understood as the theory of Hecke operators. Ramanujan also conjectured the third property | τ ( p ) | ≤ 2 p 11 2 {\displaystyle |\tau (p)|\leq 2p^{\frac {11}{2}}} for all primes p {\displaystyle p} , which is called the Ramanujan conjecture. Assuming the first two properties, Ramanujan noted that his conjecture is equivalent to the inequality

| τ ( n ) | ≤ d ( n ) n 11 2 {\displaystyle |\tau (n)|\leq d(n)n^{\frac {11}{2}}}

… excerpt ends here. Continue reading the full article.

Illustrations

Ramanujan tau function: Values of | τ ( n ) | {\displaystyle |\tau (n)|} for n < 16 , 000 {\displaystyle n<16,000} with a logarithmic scale. The blue line picks only the values of n {\displaystyle n} that are multiples of 121.
Values of | τ ( n ) | {\displaystyle |\tau (n)|} for n < 16 , 000 {\displaystyle n<16,000} with a logarithmic scale. The blue line picks only the values of n {\displaystyle n} that are multiples of 121.

Worked examples

Example 1 — a first encounter with Ramanujan tau function

Start with the simplest possible case. Write down what Ramanujan tau 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 Ramanujan tau 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 Ramanujan tau 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 Ramanujan tau function

In research
Ramanujan tau 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 Ramanujan tau 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
Ramanujan tau function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Modular forms, Multiplicative functions, Srinivasa Ramanujan, so understanding it makes those chapters shorter.
In everyday life
Look for Ramanujan tau 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.
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How to study Ramanujan tau function in 20 minutes

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

Frequently asked questions

What is Ramanujan tau function in simple terms?

In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z} } defined by ∑ n = 1 ∞ τ ( n ) q n = q ∏ n = 1 ∞ ( 1 − q n ) 24 = q ϕ ( q ) 24 = η ( z ) 24 = Δ ( z ) , {\displaystyle \sum _{n=1}^{\infty }\tau (n)…

Why does Ramanujan tau 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 Ramanujan tau 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 Ramanujan tau function.

Tags

  • Modular forms
  • Multiplicative functions
  • Srinivasa Ramanujan
  • Zeta and L-functions

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