Product of Sigma and Euler Phi Functions
Theorem
Let $n$ be an integer such that $n \ge 2$.
Let the prime decomposition of $n$ be:
- $n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$
Let $\sigma \left({n}\right)$ be the sigma function of $n$.
Let $\phi \left({n}\right)$ be the Euler phi function of $n$.
Then:
- $\displaystyle \sigma \left({n}\right) \phi \left({n}\right) = n^2 \prod_{1 \mathop \le i \mathop \le r} \left({1 - \frac 1 {p_i^{k_i + 1}}}\right)$
Proof
Better to start with $\displaystyle \phi \left({n}\right) = n \prod_{p \mathrel \backslash n} \left({1 - \frac 1 p}\right)$ and then a lot of the gnarliness goes away
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From Euler Phi Function of Integer:
- $\displaystyle \phi \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i - 1} \left({p_i - 1}\right)$
From Sigma Function of Integer:
- $\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
So:
- $\displaystyle \sigma \left({n}\right) \phi \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \left({\frac {p_i^{k_i + 1} - 1} {p_i - 1}}\right) p_i^{k_i - 1} \left({p_i - 1}\right)$
Taking a general factor of this product:
| \(\displaystyle \left({\frac {p_i^{k_i + 1} - 1} {p_i - 1} }\right) p_i^{k_i - 1} \left({p_i - 1}\right)\) | \(=\) | \(\displaystyle \left({p_i^{k_i + 1} - 1}\right) p_i^{k_i - 1}\) | $\quad$ cancelling $p_i - 1$ top and bottom | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle p_i^{2 k_i} - p_i^{k_i - 1}\) | $\quad$ multiplying out the bracket | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle p_i^{2 k_i} \left({1 - \frac 1 {p_i^{k_i + 1} } }\right)\) | $\quad$ extracting $p_i^{2 k_i}$ as a factor | $\quad$ |
So:
- $\displaystyle \sigma \left({n}\right) \phi \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} p_i^{2 k_i} \left({1 - \frac 1 {p_i^{k_i + 1} } }\right)$
Hence:
- $\displaystyle \prod_{1 \mathop \le i \mathop \le r} p_i^{2 k_i} = \left({\prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}}\right)^2 = n^2$
and the result follows.
$\blacksquare$
