I would like to know if there is any formula for

$ \prod_{x<p\leq y}\left(1+\frac1p\right) $ in terms of $\theta$ or $\psi$ functions $ \theta(x)=\sum_{p\leq x}\log p $ and $ \psi(x)=\sum_{p^\nu\leq x}\log p. $

More precisely, I need to know if we can write that product as some thing similar to the following

$ \frac{\log\theta(y)}{\log\theta(x)}+\epsilon(x,y) $

or

$ \frac{\log\theta(y+\epsilon_1(x,y))}{\log\theta(x+\epsilon_2(x,y))} $

which is equality or very sharp inequality.

avoiding the terms include

$ \frac{\log y}{\log x} $

thanks