Prime divisor count 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).
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Definition

Let $n$ be a natural number. The prime divisor count function of $n$, denoted $\omega (n)$, is defined as the number of prime divisors of $n$.

Relation with other arithmetic functions

• Mobius function: This is defined as $(-1{)}^{\omega (n)}$ for $n$ a square-free number, and is $0$ otherwise.
• Divisor count function: This is denoted $\tau (n)$, and is defined as the total number of positive divisors of $n$.
• Largest prime power divisor: Denoted $q(n)$, this is defined as the largest prime power dividing $n$. We have:

$\omega (n)\log (q(n))\ge \log (n)$.