Prove/disprove the following inequalities.

If \(M\) and \(k\) are natural numbers, then

\[1) \sum_{n=1}^{M} \phi(n) \geq \frac{1}{\sum_{n=1}^{M} \phi (n)}\]

\[2)\sum_{n=1}^{M} \phi(n) \geq \sum_{n=1}^{M} \phi(\phi(n))\]

**Notation**: \(\phi(\cdot) \) denotes Euler's totient function.

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## Comments

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TopNewestLemma:\(\phi(n^k)≥\phi(n)^k\)

proof:\(\phi(n^k)=n^{k-1}\phi(n)≥\phi^k(n)\to n^{k-1}≥\phi^{k-1}(n)\to n≥\phi(n)\) which we know is true. the result follows

3.\(n≥\phi(n)\) let \(n=\phi(x)\) the \(\phi(x)≥\phi(\phi(x))\) and the result follows. – Aareyan Manzoor · 1 year, 4 months ago

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Thank god I started from the bottom, otherwise I would've worked harder, only to receive nothing :P – Mehul Arora · 1 year, 4 months ago

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– Otto Bretscher · 1 year, 4 months ago

right, exactly. The only non-trivial one is \(\phi(n^k)=n^{k-1}\phi(n)\)Log in to reply

Q3. We will prove that \(\phi (n) > \phi (\phi (n))\)

\(\phi (n) = n \times (1- \dfrac {1}{a_1}) \times (1- \dfrac {1}{a_2}) \times \cdots \times \dfrac {1}{a_k}\) where \(n = a_1 ^{p_1} a_2 ^{p_2}\cdots a_k ^{p_k}\) where \(a_1,a_2...., a_k\) are prime numbers, and \(p_1,p_2..., p_k\) are positive integers.

Now, \(\phi (\phi (n)) = \phi (n \times (1- \dfrac {1}{a_1}) \times (1- \dfrac {1}{a_2}) \times \cdots \times \dfrac {1}{a_k}))\)

We observe that \(\phi (n) < n\) , because it is multiplied by fractions less than 1.

Thus , we show that \(\phi (\phi (n)) \leq \phi (n)\) , thus the inequality holds true. – Mehul Arora · 1 year, 4 months ago

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– Chinmay Sangawadekar · 1 year, 4 months ago

Nicely done Mehul !Log in to reply

I was about to write the other proofs as well, but Aareyan beat me ;) – Mehul Arora · 1 year, 4 months ago

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– Chinmay Sangawadekar · 1 year, 4 months ago

oh , want some more ? actually I didn't know that 2nd one was a lemma , ....Log in to reply

– Otto Bretscher · 1 year, 4 months ago

Small correction : \(\phi(n)\leq n\) since you have equality for \(n=1\)Log in to reply

– Mehul Arora · 1 year, 4 months ago

Oops yeah, thanks :DLog in to reply

I thought about them during my morning shower, and they all appear to be true. Want any proofs? They are all one liners... – Otto Bretscher · 1 year, 4 months ago

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– Chinmay Sangawadekar · 1 year, 4 months ago

Morning shower is the best place to think abt maths , ;) , Actually I thought abt them and their proofs , in my morning showerLog in to reply

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I have* :P – Mehul Arora · 1 year, 4 months ago

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