-1

I need help with a few problems of my Maths homework:

(1). Proof that arithmetic mean of $\sigma(n)$ and $\phi(n)$ is $\ge$ $n$.

(2). Proof that harmonic mean of $\sigma(n)$ and $\phi(n)$ is $\le$ $n$.

Mayank

Mayank Shekhar
  • 1
  • 1
    You should tell us what you've tried. Although, you didn't say what $\varphi$ and $\sigma$ are. – Rodrigo Dias Oct 28 '16 at 19:18
  • You need not mention your name in the question. It is displayed everytime you post anything on this site as per your profile details. –  Oct 28 '16 at 19:22
  • Fun fact: geometric mean of $\sigma(n)$ and $\varphi(n)$ is also $\le n$. An unexpected case of AM-GM, so to say. – Ivan Neretin Oct 28 '16 at 19:38

1 Answers1

0

Start with the definitions of $\sigma$ and $\phi$: we have $$ \sigma(n) = \sum_{d \vert n}{d},\quad \phi(n) = \prod_i{(p_i - 1) \cdot p_i^{e_i-1}}. $$

Expand the product and rewrite it as $ \phi(n) = \sum_{d \vert n}{a_d \cdot d}$ with coefficients $a_d \in \{-1,0,1\}$. Then we have $ \sigma(n) + \phi(n) = \sum_{d \vert n}{(a_d + 1) \cdot d} $, and since $a_n = 1$: $$ \sigma(n) + \phi(n) = 2 \cdot n + \sum_{d \vert n, d < n}{(a_d + 1) \cdot d} \ge 2n. $$ This shows that the arithmetic mean is $\ge n$. A similar computation shows the second claim.

Dietrich Burde
  • 140,055