View source for Number of divisors

{{TEX|done}}{{MSC|11A25}}

''of a natural number $n$''

The number of natural divisors of the number $n$. This arithmetic function is denoted by $\tau(n)$ or $d(n)$. The following formula holds:
$$
\tau(n) = (a_1+1) \cdots (a_k+1)
$$ 
where
$$
n = p_1^{a_1} \cdots p_k^{a_k}
$$
is the canonical expansion of $n$ into prime power factors. For prime numbers $p$, $\tau(p) = 2$, but there exists an infinite sequence of $n$ for which
$$
\tau(n) \ge 2^{1-\epsilon} \frac{\log n}{\log\log n}\,,\ \ \epsilon > 0 \ .
$$

On the other hand, for all $\epsilon > 0$,
$$
\tau(n) = O(n^\epsilon)\ .
$$

$\tau$ is a [[multiplicative arithmetic function]] and is equal to the number of points with natural coordinates on the hyperbola $xy = n$. The average value of $\tau(n)$ is given by Dirichlet's asymptotic formula (cf. [[Divisor problems]]). 

The function $\tau_k(n)$, which is the number of solutions of the equation $n = x_1\cdots x_k$ in natural numbers $x_1,\ldots,x_k$, is a generalization of the function $\tau$.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,   "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  K. Prachar,   "Primzahlverteilung" , Springer  (1957)</TD></TR>
</table>



====Comments====
The [[Average order of an arithmetic function|average value]] of the number of divisors was obtained by P. Dirichlet in 1849, in the form
$$
\sum_{n \le x} \tau(n) = x \log x + (2 \gamma - 1)x + O(\sqrt x) \ .
$$

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapt. XVI</TD></TR>
</table>