I don't know why but i assumed that $\ln \theta(x) + E < \sum \limits_{p \leq x} \frac{\ln p}{p} $ for all real numbers $x \geq 2$ and with $E = -\gamma - \sum \limits_{P} \frac{\ln p}{p(p-1)} \approx -1.33258$ is equivalent to RH, i saw it somewhere
When i went back to find it in the articles in my laptop I just found the one directly related to Robin's inequality $\ln \ln \theta(x) +B < \sum \limits_{p \leq x} \frac{1}{p} $ for all $x\geq 3$.
Is there a ref talking about the first inequality being equivalent to RH, or a direct proof of it from Robin,Nicolas inequality ?
Thanks.
Define von Mangold function as $\Lambda(n)=\log p[n=p^k]$ then
$$ \sum_{n\le x}{\Lambda(n)\over n}<\sum_{p\le x}\log p\sum_{k\ge1}p^{-k}=\sum_{p\le x}{\log p\over p}+\sum_{p\le x}\log p\sum_{k\ge2}p^{-k}<\sum_{p\le x}{\log p\over p}+\sum_p{\log p\over p(p-1)} $$
As a result, we can use the definition of the constant $E$ to get
$$ \sum_{n\le x}{\Lambda(n)\over n}+\gamma+E<\sum_{p\le x}{\log p\over p} $$
If RH is equivalent to saying
$$ \log\vartheta(x)\le\sum_{n\le x}{\Lambda(n)\over n}+\gamma $$
then your statement will be another equivalent form of RH.