I was reading a journal entry on the proof of Perron's formula, and I got stuck on one of the computations. The following is the journal entry itself:
The part I have a problem with is where they make an estimate on the first sum in (3.4). How did they get $\log x/n$?
Rewrite and expand the Dirichlet series. $$ f(s)x^s = \sum \frac{a_n}{n^s} x^s = \sum a_n \left( \frac{x}{n} \right)^s,$$ so now $\frac{x}{n}$ plays the role of $y$ in the integral. You had $\log y$ before. You you have $\log x/n$.