τ function


The τ functionMathworldPlanetmath, also called the divisor functionDlmfDlmfMathworldPlanetmath, takes a positive integer as its input and gives the number of positive divisorsMathworldPlanetmathPlanetmath of its input as its output. For example, since 1, 2, and 4 are all of the positive divisors of 4, we have τ(4)=3. As another example, since 1, 2, 5, and 10 are all of the positive divisors of 10, we have τ(10)=4.

The τ function behaves according to the following two rules:

1. If p is a prime and k is a nonnegative integer, then τ(pk)=k+1.

2. If gcd(a,b)=1, then τ(ab)=τ(a)τ(b).

Because these two rules hold for the τ function, it is a multiplicative functionMathworldPlanetmath.

Note that these rules work for the previous two examples. Since 2 is prime, we have τ(4)=τ(22)=2+1=3. Since 2 and 5 are distinct primes, we have τ(10)=τ(25)=τ(2)τ(5)=(1+1)(1+1)=4.

If n is a positive integer, the number of prime factorsMathworldPlanetmathPlanetmath (http://planetmath.org/UFD) of xn-1 over [x] is τ(n). For example, x9-1=(x3-1)(x6+x3+1)=(x-1)(x2+x+1)(x6+x3+1) and τ(9)=3.

The τ function is extremely useful for studying cyclic rings.

The sequence {τ(n)} appears in the OEIS as sequence http://www.research.att.com/ njas/sequences/A000005A000005.

Title τ function
Canonical name tauFunction
Date of creation 2013-03-22 13:30:16
Last modified on 2013-03-22 13:30:16
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 21
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11A25
Synonym divisor function
Related topic Divisor
Related topic DirichletHyperbolaMethod
Related topic 2omeganLeTaunLe2Omegan
Related topic Divisibility
Related topic ValuesOfNForWhichVarphintaun
Related topic LambertSeries
Related topic ParityOfTauFunction