τ function
The τ function, also called the divisor function
, takes a positive integer as its input and gives the number of positive divisors
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 function.
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(2⋅5)=τ(2)τ(5)=(1+1)(1+1)=4.
If n is a positive integer, the number of prime factors (http://planetmath.org/UFD) of xn-1 over ℚ[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 |