I'm beginning analytic number theory and I see this formula in Apostol's book : If $f$ has a continuous derivative $f'$ on the interval $[y,x]$, where $0 < y < x$, then
$$
\sum_{y < n \le x} f(n) = \int_y^x f(t) \, dt + \int_y^x (t-[t]) f'(t) \, dt + f(x) ([x] – x) – f(y) ([y]-y).
$$
The proof in Apostol's can be followed easily if one uses Riemann Integration. But since I meet with number theorists often I see more this kind of notation :
$$
\sum_{y < n \le x} f(n) = \int_y^x f(t) d[t] = \text{something here I don't recall} – \int_y^x [t]f'(t) dt
$$
because for some reason they can "integrate $d[t]$ and it gives $[t]$", which I don't understand, and I also don't really understand precisely what $d[t]$ stands for. I have done a measure theory course ; what I'm saying is that I don't understand all the details ; I understand that they "integrate by parts with the measure $d[t]$" which makes the proof quite simpler, but I don't understand the assumptions they make and how the details work out. I think that $d[t]$ could be a measure such that for $E \subseteq \mathbb R$ or $\mathbb C$, $d[t](E) = | \mathbb N \cap E |$, but I'm not sure.
Here's what I'm looking for : I don't want an intuitive point of view with plots or summations ; I want a formal proof from the viewpoint of a measure theorist, with details. Is there anyway this can be made clear? The reason why I want this is because I don't have much faith in the "integration by parts with $d[t]$" version of the proof, but number theorists seem to love it so much and they all sketch it ; I never managed to do it myself formally, even though I did a measure theory course.
Thanks for the help,
Best Answer
Find the book Montgomery and Vaughn's "Multiplicative Number Theory I. Classical Theory". This book is an excellent reference for many different subjects analytic number theory.
Appendix A, "The Riemann Siteltjes integral," deals with precisely your question. It is $8$ pages long, and should answer everything.
Edit: Zev kindly added links to Appendix A, so no need to check your library, or ask your prof.