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So, while playing around with the graph of Euler's totient function, it's graph was looking it could be bound between two straight lines, now the upper one was obvious to be $y=x$ or $y=x-1$ if we are willing to exclude $\phi(1)$, but the lower bound for a straight line passing through origin is some random slope around $\dfrac{16}{77}$ by seeing $\phi(2310)=480$ which seems to fit until the first $10,000$ numbers. My question is, is it even possible to find the optimum lower bound $y=ax+b$ for the Euler's totient function?

enter image description here Link for desmos graph https://www.desmos.com/calculator/kxwdny3urj

lilychou
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The Euler function is not bounded below by any function $cx - b$ with $c>0$. But it decays quite slowly, see Is the Euler phi function bounded below?, the smallest values of $\phi(n)/n$ are only about as small as $1/\log \log n$.

Erick Wong
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