Ramanujan tau function

The Ramanujan tau functionDlmfPlanetmath is the arithmetic functionMathworldPlanetmath τ: such that, for all q with |q|<1,


Thus, the Ramanujan tau function is the generating function for the Weierstrass Δ functionMathworldPlanetmath (http://planetmath.org/ModularForms).

Determining values of the Ramanujan tau function directly can be somewhat involved. For example, the values of τ(1), τ(2), and τ(3) will be determined:

To determine τ(1), τ(2), and τ(3), we need to find the coefficient of q, q2, and q3, respectively, of the expression


Note that we only need to consider k=1 and k=2, since higher values of k yield powers (http://planetmath.org/Power) of q that are too large. Thus:

q(1-q)24(1-q2)24 =q(1-24q+276q2-)(1-24q2+)

Hence, τ(1)=1, τ(2)=-24, and τ(3)=252.

The sequence {τ(n)} appears in the OEIS as sequence http://www.research.att.com/ njas/sequences/A000594A000594.

Although the values of |τ(n)| seem to increase rapidly as n increases, the conjecture that τ(n)0 for all n has not yet been proven. This conjecture is known as Lehmer’s conjecture.

The Ramanujan tau function has the following properties:

  • It is a multiplicative functionMathworldPlanetmath: For a,b with gcd(a,b)=1, we have τ(ab)=τ(a)τ(b).

  • For any prime p and any n,

  • For any prime p,


Ramanujan asserted that τ several congruencesMathworldPlanetmathPlanetmath, all of which have been proven. Some simpler examples of such congruences include:

  • For any n,

  • For any n and for any nonnegative integer r<7 which is a quadratic residueMathworldPlanetmath modulo 7,

  • For any n and for any nonnegative integer r<23 which is a quadratic residue modulo 23,



Title Ramanujan tau function
Canonical name RamanujanTauFunction
Date of creation 2013-03-22 17:51:24
Last modified on 2013-03-22 17:51:24
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 12
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11F11
Classification msc 11A25
Synonym Ramanujan’s tau function
Related topic ModularForms
Related topic ModularDiscriminant
Related topic Ramanujan
Related topic ApplicationsOfSecondOrderRecurrenceRelationFormula
Defines Lehmer’s conjecture