# All ones function

From Number

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 be a commutative unital ring. The **all ones function** from the natural numbers to is the function that sends every natural number to the identity element of .

The all ones function is typically denoted by the letter .

## Relation with other arithmetic functions

- The inverse of this function with respect to the Dirichlet product is the Mobius function .
- The square of this function with respect to the Dirichlet product is the divisor count function.
- Taking the Dirichlet product of a function with this function is equivalent to summing up over all the positive divisors:

.