# Number of divisors

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2010 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

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 $\tau$.

How to Cite This Entry:
Number of divisors. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Number_of_divisors&oldid=51004
This article was adapted from an original article by N.I. Klimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article