# Number of divisors

From Encyclopedia of Mathematics

A function of a natural argument, , equal to the number of natural divisors of the number . This arithmetic function is denoted by or . The following formula holds:

where

is the canonical expansion of into prime factors. For prime numbers , , but there exists an infinite sequence of for which

On the other hand, for all ,

is a multiplicative arithmetic function and is equal to the number of points with natural coordinates on the hyperbola . The average value of is given by Dirichlet's asymptotic formula (cf. Divisor problems). The function , which is the number of solutions of the equation in natural numbers , is a generalization of the function .

#### References

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

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

#### Comments

#### 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=18293

This article was adapted from an original article by N.I. Klimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article