📂Number Theory

In Analytic Number Theory

Definition ¹

The arithmetic function defined as follows $u$ is called the unit function. $$ u(n) := 1 $$

Basic Properties

• [1] Unit series: Equals the number of divisors $\sigma_{0}$. In other words, $$ \sum_{d \mid n} u(d) = \sigma_{0} (n) $$
• [2] Completely multiplicative: For all $m,n \in \mathbb{N}$, $u(mn) = u(m) u(n)$

Explanation

$$ \begin{matrix} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ u (n) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \sum_{d \mid n} u(d) & 1 & 2 & 2 & 3 & 2 & 4 & 2 & 4 & 3 & 4 \end{matrix} $$ As can be inferred from the name “unit function”, it is a very important function. Considering convolution, the series $F$ of any arithmetic function $f$ is actually expressed as follows. $$ f \ast\ u = F $$

Proof

[1]

$$ \sum_{d \mid n} u(d) = \sum_{d \mid n} 1 = \sigma_{0} (n) $$ Trivial due to the definition of the divisor function.

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[2]

$$ u(mn) = 1 = 1 \cdot 1 = u(m) u(n) $$

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