# How do I write $\sum\limits_{p \leq q \text{ prime}}\log (p-x)$ using Chebyshev's function(s)?

Question: If $$x\in \mathbb{N}$$ and $$p$$ and $$q$$ are prime numbers how do I write $$\sum\limits_{p \leq q \text{ prime}}\log (p-x)$$ using Chebyshev's function(s) ?

I have some partial results: The case $$x=0$$ is answered explicitly by definition of the Chebyshev function. The case $$x=1$$ can be solved "explicitly" and approximately via Merten's third theorem$$^1$$. (See a related problem and answer$$^2$$). In particular for $$q$$ large

$$\sum\limits_{p \leq q \text{ prime}}\log{(p-1)}\sim\vartheta(q-1)+\log{\left(\frac{q}{\log{q}}\right)} -\gamma$$ ; where $$\gamma$$ is the Euler-Mascheroni constant$$^3$$. I think it is also fair to ask if there are functions other than Chebyshev that are better suited for this calculation.