Are these equalities wrong $\sum_{n\le x}\mu(n)o(\frac xn)=o(x\sum_{n\le x}\frac 1n)=o (x\log x) $?

Question

I simply found an asymptotic relation for $\sum_{n\le x}\mu(n)o(\frac xn)$ like below:

$$\sum_{n\le x}\mu(n)o(\frac xn)=o(x\sum_{n\le x}\frac 1n)=o (x\log x) $$

But Basil Gordon, an American mathematician, in his thesis found that like below:

$$\sum_{n\le x}\mu (n)o (\frac xn)=\sum_{n\le\frac x{\log x}}\mu (n)o (\frac xn)+\sum_{\frac x{\log x}\le n \le x}\mu (n)O (\frac xn) $$

Then since $|\mu (n)|\le 1$

$$o (\sum_{n\le\frac x{\log x}}\frac xn)=o (x\log x) $$

and

$$O (\sum_{\frac x {\log x}\le n\le x}\frac xn)=O (x\log x-x\log\frac x{\log x})=O (x\log\log x)=o (x\log x) $$

Could anyone explain for me why Basil Gordon has done that like this? Is my answer wrong?

Answer 1

Use small o notation with a special care on which variable goes to infinity. What you said is basically the following.

Claim(?): Let $f(n,x)$ be a function in $n$ and $x$. Suppose $\lim \frac{f(n,x)}{x/n} = 0$. Then we have $\lim \frac{\sum_{n\leq x} f(n,x)}{\sum_{n \leq x} x/n} = 0$.

Is this claim true? It depends on context (so I intentionally omit how the limit goes). What you wrote in your question, you interpreted small o notation by $\lim_{n\leq x, x\rightarrow\infty} f(n,x) = 0 $ uniformly in $n\leq x$.

I haven't looked into his thesis, but based on what was written, I assume what the author meant by the formula $o(x/n)$ is any function of the form $\phi(x/n)$ where $\phi$ is a function which satisfies $\lim_{u\rightarrow \infty} \phi(u) / u = 0$. (or, $o(x/n)$ uniformly as $x/n \rightarrow \infty$). What makes you more confusing is that, in the last formula $o(x log x)$, his limit was taken as $x \rightarrow \infty$. So it is indeed the same notation with different meaning from line to line. So, I'll write $o_{x/n}$ to denote the former and $o_x$ for the latter. Then your interpretation is $$ \sum_{n\leq x} o_x \left ( \frac{x}{n} \right ) = x o_x \left (\sum_{n\leq x} \frac{1}{n} \right ) = o_x (x \log x),$$ which is obviously true. I believe the author's intention was $$ \sum_{n\leq x} o_{x/n} (x/n) = o_x ( x \log x),$$ which is not automatic(for instance, when $x/2\leq n\leq x$, $x/n$ is bounded so we don't know if it is $o_x(1)$. Thus the author goes in two steps to handle this.

What reuns said I believe is that $\sum_{n\leq x} o_n (1/n^2) \neq o_x(1)$. So it all depends on the context after all.