Trying to understand a paper which makes use of the following "fact" about logarithms. Can someone please explain why it is so?
Let $\phi(x) = \sum_{x/a \leq n\leq x/b} 1/n$. Then $\lim_{x\to \infty}\phi(x) = \log(a/b)$. In other words, $$\log(a/b) = \lim_{x\to \infty}\sum_{x/a \leq n\leq x/b}\frac{1}{n}.$$
It is known that $$\lim_{n\to\infty}\left(\sum_{i=1}^n\frac1i-\ln n\right)=\gamma$$ Therefore $$\lim_{x\to\infty}\sum_{x/a\le n\le x/b}\frac 1n=\lim_{x\to\infty}\left(\sum_{i=1}^{\lfloor x/b\rfloor}\frac1i-\sum_{i=1}^{\lceil x/a\rceil}\frac1i\right)=\lim_{x\to\infty}(\ln\lfloor x/b\rfloor+\gamma-\ln\lceil x/a\rceil-\gamma)$$ $$=\lim_{x\to\infty}\ln\frac{\lfloor x/b\rfloor}{\lceil x/a\rceil}=\ln\frac ab$$