Theorem 3.2 of Apostol confusion!!

Question

I was working on Introduction to Analytic number theory while I found something really confusing due to my limited IQ. He gives a theorem which I paste below

And theorem 3.2 is an application of theorem 3.1.

As you can see, I mark +1 in the picture. My question is where does this 1 come from. I thought we just take y=1 in Euler's summation formula then we should have $f(1)([1]-1)=0$.

Answer 1

Euler summation formula has $$\sum_{y < n \le x} \frac1n$$ where $y=1$. But below in the picture the author writes $$\sum_{n \le x}\frac1n$$ means $n$ starts from $1$. So $\dots$

Answer 2

Without that term you will get sum of $\frac 1 n$ over $1<y\leq x$. Note the strict inequality here. You have add the first term corresponding to $n=1$ to get the sum over $n \leq x$.