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Suppose $p_n$ is the $n$-th prime number. Does $\sum\limits_{n=1}^\infty \ln\left(\frac{p_n}{p_n - 1}\right)$ converge?

Where did this question arise from: I was trying to find $\inf_{n \in \mathbb{N}} \frac{\phi (n)}{n}$, where $\phi$ is Euler totient function. If $p$ is prime, then $\frac{\phi (p^n)}{p^n} = \frac{p - 1}{p}$. As $\phi$ is multiplicative, $\phi(n) = \prod\limits_{p\mid n; p \text{ is prime}} \frac{p - 1}{p}$. That means, that $ \inf_{n \in \mathbb{N}} \frac{\phi (n)}{n} = \prod\limits_{n = 1}^\infty \frac{p - 1}{p}$. And that results in $\inf_{n \in \mathbb{N}} \frac{\phi (n)}{n} = 0$ iff $\sum\limits_{n=1}^\infty \ln\left(\frac{p_n}{p_n - 1}\right)$ diverges.

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    $\begingroup$ Yet another way of doing this : the sum (if it converges) is the logarithm of the product $\prod_n \left(1+\frac1{p_n-1}\right)$ and it's 'broadly known' that this is exactly equivalent to convergence of the series $\frac1{p_n-1}$. $\endgroup$ – Steven Stadnicki Jun 11 '18 at 18:01
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Notice that

$$\ln\left(\frac{p_n}{p_n - 1}\right) = \ln p_n - \ln (p_n - 1) \approx \frac 1 {p_n}$$

by the mean value theorem, using that $\frac{d}{dx} \ln x = \frac 1 x$.

It is well known that $\sum p_n^{-1}$ diverges.

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Observe that $$\ln\left(\frac{p_n}{p_n-1}\right)=\ln\left(1+\frac{1}{p_n-1}\right)=\frac{1}{p_n-1}+O\left(\frac{1}{(p_n-1)^2}\right).$$Thus,$$\left|\sum_{p}\ln\left(\frac{p}{p-1}\right)-\sum_p\frac{1}{p-1}\right|\ll1.$$From this, we can deduce that $\sum_p\ln\left(\frac{p}{p-1}\right)$ diverges since $\sum_p\frac{1}{p-1}$ diverges.

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Note $\ln(p/(p-1)) = \ln(1/(1 - 1/p))$, so $\sum_{p\leq x} \ln(p/(p-1)) = \ln(\prod_{p\leq x} 1/(1-1/p))$. Intuitively, $\prod_p 1/(1-1/p) = \zeta(1) = \infty$, and this calculation can be justified. Thus, letting $x\rightarrow \infty$, we get $\sum_p \ln(p/(p-1)) = \ln(\prod_p 1/(1-1/p)) = \ln(\infty) = \infty$.

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Another way, it is known that $$\ln{(1+x)}\geq \frac{x}{1+x}, \forall x>-1$$ and $$\ln{\left(\frac{p_n}{p_n-1}\right)}=\ln{\left(1+\frac{1}{p_n-1}\right)}\geq \frac{\frac{1}{p_n-1}}{1+\frac{1}{p_n-1}}=\frac{1}{p_n}$$ thus $$\sum\limits_{n}\ln{\left(\frac{p_n}{p_n-1}\right)} \geq \sum\limits_{n}\frac{1}{p_n}$$ which diverges.

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