Proofs of Asymptotics

I'm going through a few proofs to make sure I understand them and in two of the proofs, there is a step I don't understand.

1) In the first, we have that $f(x) = g(x) + ln\left(\frac{\lfloor x \rfloor}{x}\right)$. So, if $|g(x)| < \frac{2}{\lfloor x \rfloor}$, then $|f(x)| < \frac{2}{\lfloor x \rfloor} + \frac{1}{\lfloor x \rfloor}$. To me, this implies that $\left|ln\left(\frac{\lfloor x \rfloor}{x}\right)\right| < \frac{1}{\lfloor x \rfloor}$ and, unless I've gone wrong, this is not true?

2) In the second, we have that $\int_{1}^x \! \frac{ln(u)}{u} \, \mathrm{d}u = \frac{1}{2} \, ln^2(x)$. We use this on $\int_{1}^{\lfloor x \rfloor} \frac{ln(u)}{u} \, \mathrm{d}u$ and, somehow, get $\frac{1}{2} ln^2(x)$ and not $\frac{1}{2} ln^2(\lfloor x \rfloor)$. I assumed that this was because it was absorbed into O$\left(\frac{ln(x)}{x}\right)$, but I couldn't prove this.

Would anyone be able to help me here? It's to prove some asymptotic formulas for $\displaystyle\sum\limits_{n \leq x} \frac{1}{n}$ and $\displaystyle\sum\limits_{n \leq x} \frac{ln(n)}{n}$ and I'm following Shapiro's derivations.

If we write $x = k+h$ with $0 \leqslant h < 1$, then we have

$$\left\lvert \ln \frac{\lfloor x\rfloor}{x}\right\rvert = \ln \frac{k+h}{k} = \ln \left(1+\frac{h}{k}\right) \leqslant \frac{h}{k} < \frac{1}{k} = \frac{1}{\lfloor x\rfloor}.$$

In the second, we have

$$\ln x = \ln \left(\lfloor x\rfloor \frac{x}{\lfloor x\rfloor}\right) = \ln \lfloor x\rfloor + \ln \frac{x}{\lfloor x\rfloor},$$

and we saw above that the last terms is smaller than $1/\lfloor x\rfloor$, so

$$\ln^2 x < \ln^2 \lfloor x\rfloor + 2 \frac{\ln \lfloor x\rfloor}{\lfloor x\rfloor} + \frac{1}{\lfloor x\rfloor^2},$$

so the difference can be absorbed in an $O\left(\frac{\ln x}{x}\right)$ term.

• Thank you very much. I'm a bit slow, so I'm just going to go through it now and make sure that I can understand it. So, give me a second. – user60126 Feb 28 '14 at 23:33
• Sure, no problem. If something isn't clear, don't hesitate to ask. – Daniel Fischer Feb 28 '14 at 23:34
• Ok, so I have two questions. I'll understand if you want to let me ponder over them and not answer them. :D The first is why is ln(1+h/k) < (h/k)? I came up with this possibility in my notes but couldn't think of why it would be true. The second is, if we have that f(x) $- ln^2(\lfloor x \rfloor) \in$ O(ln(x)\x), then we are allowed to add two terms, take something smaller and it is still in O(ln(x)\x)? If I can get the answer to those two questions, I understand the proofs. – user60126 Feb 28 '14 at 23:49
• For the first, we have $\ln (1+x) < x$ for $x > 0$ (and since $h = 0$ is possible, that ought to have been a $\leqslant$ there). You can see that for example by the mean value theorem, $\ln (1+x) - \ln 1 = \ln'(1+\xi)\cdot x = \frac{1}{1+\xi}\cdot x$. – Daniel Fischer Feb 28 '14 at 23:54
• It depends. If you know (can prove) that $f(x) - \ln^2 x$ is always non-negative, then we don't need it. Otherwise, we can have $$\left\lvert f(x) - \ln^2 x\right\rvert > \left\lvert f(x)-\ln^2 \lfloor x\rfloor\right\rvert,$$ and then we need an estimate to show that it still has the same $O(\cdot)$-bound. – Daniel Fischer Mar 1 '14 at 11:42