# 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: 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 .

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