$\tau$ function TauFunction 2003-03-10 02:09:51 2008-05-16 12:01:30 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{function} The $\tau$ \emph{function}, also called the \emph{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 $\tau (4)=3$. As another example, since $1$, $2$, $5$, and $10$ are all of the positive divisors of $10$, we have $\tau (10)=4$. The $\tau$ function behaves according to the following two rules: 1. If $p$ is a prime and $k$ is a nonnegative integer, then $\tau(p^k)=k+1$. 2. If $\gcd(a,b)=1$, then $\tau(ab)=\tau(a)\tau(b)$. Because these two rules hold for the $\tau$ function, it is a multiplicative function. Note that these rules work for the previous two examples. Since $2$ is prime, we have $\tau(4)=\tau(2^2)=2+1=3$. Since $2$ and $5$ are distinct primes, we have $\tau(10)=\tau(2\cdot 5)=\tau(2)\tau(5)=(1+1)(1+1)=4$. If $n$ is a positive integer, the number of \PMlinkname{prime factors}{UFD} of $x^n-1$ over $\mathbb{Q}[x]$ is $\tau(n)$. For example, $x^9-1=(x^3-1)(x^6+x^3+1)=(x-1)(x^2+x+1)(x^6+x^3+1)$ and $\tau(9)=3$. The $\tau$ function is extremely useful for studying cyclic rings. The sequence $\{\tau(n)\}$ appears in the OEIS as sequence \PMlinkexternal{A000005}{http://www.research.att.com/~njas/sequences/A000005}.