All ones function

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 $\mathbb {N}$ 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 $\mu$.
• 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:

$(f*U)(n)=\sum _{d|n}f(d)$.