# Number of divisors

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 $\tau$.

#### References

[1] | I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) |

[2] | K. Prachar, "Primzahlverteilung" , Springer (1957) |

#### Comments

The 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

[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XVI |

**How to Cite This Entry:**

Number of divisors.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Number_of_divisors&oldid=51004