The asymptotic behaviour of $\sum_{1\leq k\leq N-1}\int_{p_k}^{p_{k+1}}\log x d[x]$, where $p_n$ is the nth prime number 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.
 A: We have, using integration by parts of the Riemann–Stieltjes integral, $$\int_{p_{k}}^{p_{k+1}}\log\left(x\right)d\left[x\right]=p_{k+1}\log\left(p_{k+1}\right)-p_{k}\log\left(p_{k}\right)-\int_{p_{k}}^{p_{k+1}}\frac{\left[x\right]}{x}dx
 $$ and so we can see that $$\sum_{k=1}^{N-1}\left(p_{k+1}\log\left(p_{k+1}\right)-p_{k}\log\left(p_{k}\right)\right)=p_{N}\log\left(p_{N}\right)-2\log\left(2\right).
 $$ Now let us focus on the integral. Since $\left[x\right]=x+O\left(1\right)
 $ we have $$\int_{p_{k}}^{p_{k+1}}\frac{\left[x\right]}{x}dx=p_{k+1}-p_{k}+O\left(\log\left(p_{k+1}\right)\right)
 $$ and, since from the PNT we know that $\sum_{p\leq x}\log\left(p\right)\sim x
 $ we have $$\sum_{k=1}^{N-1}\int_{p_{k}}^{p_{k+1}}\frac{\left[x\right]}{x}dx=p_{N}-2+O\left(\sum_{k=1}^{N-1}\log\left(p_{k+1}\right)\right)=O\left(p_{N}\right)
 $$ hence $$f\left(N\right)=p_{N}\log\left(p_{N}\right)+O\left(p_{N}\right)\sim N\log^{2}\left(N\right)+O\left(N\log\left(N\right)\right).
 $$ About the limit, since $t-1\leq\left[t\right]\leq t
 $ we can observe that $$p_{k+1}-p_{k}-\log\left(p_{k+1}\right)+\log\left(p_{k}\right)\leq\int_{p_{k}}^{p_{k+1}}\frac{\left[t\right]}{t}dt\leq p_{k+1}-p_{k}
 $$ hence $$p_{N}\log\left(p_{N}\right)-2\log\left(2\right)-p_{N}+2\leq f\left(N\right)\leq p_{N}\log\left(p_{N}\right)-2\log\left(2\right)-p_{N}+2+\log\left(\frac{p_{N}}{2}\right)$$ and using the bounds $$\log\left(N\right)+\log\left(\log\left(N\right)\right)-1\leq\frac{p_{N}}{N}\leq\log\left(N\right)+\log\left(\log\left(N\right)\right),\, N\geq6
 $$ we can see that $$\lim_{N\rightarrow\infty}f\left(N+1\right)-f\left(N\right)=\infty.$$
