All ones function - Revision history

Vipul: Created page with '{{arithmetic function}} ==Definition== Let $R$ be a commutative unital ring. The '''all ones function''' from the natural numbers $\mathbb{N}$ to
2009-04-29T03:36:04Z

Created page with '{{arithmetic function}} ==Definition== Let <math>R</math> be a commutative unital ring. The '''all ones function''' from the natural numbers <math>\mathbb{N}</math> to <mat...'

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{{arithmetic function}}

==Definition==

Let <math>R</math> be a [[commutative unital ring]]. The '''all ones function''' from the natural numbers <math>\mathbb{N}</math> to <math>R</math> is the function that sends every natural number to the identity element <math>1</math> of <math>R</math>.

The all ones function is typically denoted by the letter <math>U</math>.

==Relation with other arithmetic functions==

* The inverse of this function with respect to the [[Dirichlet product]] is the [[Mobius function]] <math>\mu</math>.
* The square of this function with respect to the [[Dirichlet product]] is the [[divisor count function]].
* Taking the Dirichlet product of a function <math>f</math> with this function is equivalent to summing up <math>f</math> over all the positive divisors:

<math>(f * U)(n) = \sum_{d|n} f(d)</math>.