The Euler function

The Euler function $\phi (n)$ is defined as the number of positive integers not exceeding $n$ and coprime with $n$. Also, let $N_k$ be the product of the first $k$ prime numbers, and $\gamma$ be Euler's constant.

Then RH is equivalent to each of the following statements :

$$\frac{N_k}{\phi(N_k)} > e^{\gamma} \log \log N_k,$$

for all $k$'s ;

$$\frac{N_k}{\phi(N_k)} > e^{\gamma} \log \log N_k,$$

for all but finitely many $k$'s.

This is due to Nicolas [ MR 85h:11053].

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