Let $p_k$ is the kth prime number and consider for $N\geq 2$ the arithmetic function $$f(N)=\sum_{k=1}^{N-1}\int_{p_k}^{p_{k+1}}\log(x) d[x]$$ where $[x]$ is the integer part function (provide us in your answer the definition of the function [x]). Then I believe that it could be a nice exercise
Question. a) Compute the asymptotic behaviour of $f(N)$ as $N\to \infty$ as an asymptotic equivalence $$f(N)\sim \text{something}.$$ b) Evaluate $$\lim_{N\to\infty}f(N+1)-f(N).$$
I did this exercise yesterday, computing unconditionally (it is without assumption of additional hypothesis or conjectures) with well-knonw asymptotics. Can your repeat the computations to see if my aproach was right? Many thanks.