📂Number Theory

Analytic Number Theory: Norms

Definition ¹

The arithmetic function defined as below $N$ is called a norm. $$ N(n) := n $$

Basic Properties

• [1] Norm Series: Sigma function $\sigma = \sigma_{1}$. In other words, $$ \sum_{d \mid n } N(d) = \sigma_{1}(n) $$
• [2] Complete Multiplicativity: For all $m,n \in \mathbb{N}$, $N(mn) = N(m) N (n)$

Explanation

$$ \begin{matrix} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ N(n) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \sum_{d \mid n} N(d) & 1 & 3 & 4 & 7 & 6 & 6 & 8 & 15 & 13 & 18 \end{matrix} $$ The reason this seemingly ordinary function is called a norm, is because it represents the size of a given number, similar to the norm of Gaussian rings or the norm of Eisenstein rings. However, despite such a naming, $N$ is defined as an arithmetic function, so it is not a norm in the general sense of the word, which is important to note.

Proof

[1]

$$ \sigma_{\alpha} (n) := \sum_{d \mid n} d^{\alpha} $$

$$ \sum_{d \mid n } N(d) = \sum_{d \mid n } d = \sum_{d \mid n } d^{1} = \sigma_{1}(n) $$

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[2]

$$ N(mn) = mn = N(m) N(n) $$

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