Is there anything known in general about upper and lower bounds for $\prod_{i\leq n\vee p_n>k}(p_i-k)$

Question

I have no specific reason to ask this question other than seeing that it comes up quite often when I'm playing around with prime numbers. Let $$f(n,k)=\prod_{i \leq n\vee p_n>k}(p_i-k)$$ Where $p_i$ is the $i$'th prime. I was wondering whether there are good lower and upper bounds for this function. From here, I know that: $$f(n,0)<n(1+\dfrac{1}{2\log n})$$ After some testing it looks like $f(n,k)\sim\log p_n\cdot f(n,0)$ But I can't prove anything.

Answer 1

Let $m = \pi(k)$. Then for $n > m$, $$\log f(n,k) - \log f(n,0) = \sum_{i=m+1}^n \log(1 - k/p_i) - \sum_{i=1}^m \log(p_i) $$ As $i \to \infty$, $\log(1 - k/p_i) \sim \dfrac{-k}{p_i} \sim \dfrac{-k}{i \log i}$, and $\int_{2}^n 1/(t \log t)\; dt \sim \log \log n$, so we should get $$ \log f(n,k) - \log f(n,0) \sim -k \log \log n$$ and thus $$\dfrac{f(n,k)}{f(n,0)} = (\log n)^{-k + o(1)}$$ I think that $o(1)$ can be improved to $O(1/\log n)$.