Divisor sum function: Difference between revisions

Revision as of 23:13, 21 March 2009

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
View a complete list of arithmetic functions

Definition

Let $n$ be a natural number. The divisor sum function of $n$ , denoted $\sigma (n)$ , is defined in the following equivalent ways:

$\sigma$ is the Dirichlet product of the identity function on the natural numbers and the all-one function: the function sending every natural number to $1$ .
We have $\sigma (n)=\sum _{d|n}d$ .

$\sigma$ is a multiplicative function but not a completely multiplicative function.

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 Let <math>n</math> be a [[natural number]]. The '''divisor sum function''' of <math>n</math>, denoted <math>\sigma(n)</math>, is defined in the following equivalent ways:
-# <math>\sigma(n) = \sum_{d|n} d</math>.
 # <math>\sigma</math> is the Dirichlet product of the [[identity function]] on the natural numbers and the [[all-one function]]: the function sending every natural number to <math>1</math>.
+# We have <math>\sigma(n) = \sum_{d|n} d</math>.
 <math>\sigma</math> is a [[multiplicative function]] but not a [[completely multiplicative function]].