All Questions

Is the inequality $\prod \limits_{p \leq \sqrt{x}} (1+\frac{1}{p^2-1}) \prod \limits_{p \leq x} (1+\frac{1}{p}) \leq e^\gamma \ln(\theta(\sqrt{x})+\theta(x))$ where $\theta(x)$ is the Chebyshev's ...

Are there numbers $k > 1$ and $c > 1$ such that: 1 ) $\theta(c) \geq c \left( 1-\frac{1}{5 \ln^2(c)} \right) $ 2 ) $\frac{c}{1+\frac{1}{\ln^4(c)}} \leq p(\pi(c))$ where $p(n)$ is the $n$-th ...

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ...