Number of divisors

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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:


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 .


[1] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[2] K. Prachar, "Primzahlverteilung" , Springer (1957)



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