数学 – 整数 - 2つの整数論的関数(2つの乗法的関数)

代数系入門
(岩波書店)
松坂 和夫 (著)

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代数系入門 (松坂 和夫(著)、岩波書店)の第1章(整数)、8(2つの整数論的関数)、問6-b.を解いてみる。

問6-b.

$\begin{array}{l}{\theta }_1\left(1\right)=1\\ \left(a,b\right)\Rightarrow {\theta }_1\left(ab\right)={\theta }_1\left(a\right){\theta }_1\left(b\right)\\ \\ {\theta }_2\left(1\right)=1\\ \left(a,b\right)\Rightarrow {\theta }_2\left(ab\right)={\theta }_2\left(a\right){\theta }_2\left(b\right)\\ \\ \theta \left(1\right)={\theta }_1\left(1\right){\theta }_2\left(1\right)\\ =1·1\\ =1\\ \\ \left(a,b\right)\Rightarrow \\ \theta \left(ab\right)={\theta }_1\left(ab\right){\theta }_2\left(ab\right)\\ ={\theta }_1\left(a\right){\theta }_1\left(b\right){\theta }_2\left(a\right){\theta }_2\left(b\right)\\ ={\theta }_1\left(a\right){\theta }_2\left(a\right){\theta }_1\left(b\right){\theta }_2\left(b\right)\\ =\theta \left(a\right)\theta \left(b\right)\end{array}$