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 R be a commutative unital ring. The all ones function from the natural numbers 
to R is the function that sends every natural number to the identity element 1 of R.
The all ones function is typically denoted by the letter U.
Relation with other arithmetic functions
- The inverse of this function with respect to the Dirichlet product is the Mobius function μ.
- The square of this function with respect to the Dirichlet product is the divisor count function.
- Taking the Dirichlet product of a function f with this function is equivalent to summing up f over all the positive divisors:
| (f * U)(n) = | ∑ | f(d) |
| d | n |
.
