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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.

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