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There is a well-known question that seeks the asymptotic behaviour of this function, for $x\geq 2$: $$\sum_{n\leq x} \frac{\phi(n)}{n^2}.$$ See, for example, Apostol "Introduction to Analytic Number Theory" question 6 on page 71.

I have found three posts on stackexchange related to this question, namely post A, post B, and post C. Using methods available to someone who has just read Chapter 3 of Apostol's text, the best solution to this question would seem to be apatch's response to post A.

The method boils down to showing the following identity $$\sum_{d>x}\frac{\mu(d)\log d}{d^2} = O\left(\frac{\log x}{x}\right).$$

This is where I get unstuck. The first step is clearly $$\left|\sum_{d>x}\frac{\mu(d)\log d}{d^2}\right| \leq \sum_{d>x}\frac{\log d}{d^2}$$ but then what do you do with what remains?

One solution I found simply jumps from here to $O(\frac{\log x}{x})$, which seems non-trivial. Apatch's solution in post A does the following: $$\sum_{d>x}\frac{\log d}{d^2}=\sum_{d>x}\frac{\log d}{d^\frac{1}{2}}.\frac{1}{d^\frac{3}{2}}<\frac{\log x}{x^\frac{1}{2}}\sum_{d>x}\frac{1}{d^\frac{3}{2}}$$ but I don't see how can you justify the inequality, since $\frac{\log x}{\sqrt{x}}$ only reaches its maximum until around $x\approx 7.39$, which is well above the $x>2$ condition stated in the question.

Eric Naslund's solution to post C states that $$\sum_{d> x}\frac{\mu(d)\log d}{d^2}=O(1/x)$$ but this less elementary result is presumably even harder to derive.

Can anyone identify the "correct" answer, and explain why?

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rbrignall
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2 Answers2

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Hint:

$$ \sum_{d \geq x} \frac{\log d}{d^2} \leq \frac{\log x}{x^2} + \int_x^\infty \frac{\log t}{t^2}\,dt. $$

Antonio Vargas
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  • Thanks! I'm guessing this is what Apostol intended readers to use: it is suitably elementary (and I'm now wondering why I didn't think of it). – rbrignall Nov 18 '13 at 22:35
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One way to show that

$$\sum_{n>x}\frac{\mu(n)\log n}{n^{2}}=O\left(\frac{\log x}{x}\right)$$

is to split into diadic intervals and use the bound $\sum_{n> y}\frac{1}{n^2}\ll \frac{1}{y}$. We have that

$$\sum_{n>x}\frac{\mu(n)\log n}{n^{2}}\leq\sum_{k=1}^{\infty}\sum_{2^{k-1}x<n\leq2^{k}x}\frac{\log n}{n^{2}} $$

$$\leq \sum_{k=1}^{\infty}\log(2^{k}x)\sum_{2^{k-1}x<n\leq2^{k}x}\frac{1}{n^{2}} $$

$$\ll \sum_{k=1}^{\infty}\log(2^{k}x)\frac{1}{2^{k-1}x}$$

$$=\frac{\log x}{x}\sum_{k=1}^{\infty}\frac{1}{2^{k-1}}+\frac{1}{x}\sum_{k=1}^{\infty}\frac{k\log2}{2^{k-1}}\ll\frac{\log x}{x}.$$

In my answer that you mentioned, I was thinking of this elementary bound, and forgot the $\log x$ term. However, the result I quoted is not false - we just need to take into account the cancellation resulting from the $\mu(n)$ term. Letting $M(x)=\sum_{n\leq x} \mu(n)$, integration by parts tells us that

$$\sum_{n>x}\frac{\mu(n)\log n}{n^{2}}=\int_{x}^{\infty}\frac{\log t}{t^{2}}d\left(M(t)\right)$$

$$=\frac{M(x)\log x}{x^{2}}+\int_{x}^{\infty}\frac{M(x)(2\log x-1)}{x^{3}}dx.$$ By the prime number theorem, there exists $c>0$ such that $$M(x)\ll xe^{-c\sqrt{\log x}},$$ and so we see that there is a constant $c_1>0$ such that $$\sum_{n>x}\frac{\mu(n)\log n}{n^{2}}\ll \frac{1}{x}e^{-c\sqrt{\log x}}.$$

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Eric Naslund
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  • Hi, thanks, that diadic approach is cute. Also, apologies: I didn't mean to suggest what you said in your earlier post was necessarily wrong (though I'm guessing Apostol did not have this particular argument in mind). – rbrignall Nov 18 '13 at 22:24
  • Can you give a hint or so how the prime number theorem gives the bound $O(x e^{-c\sqrt{\log x}})$ for the Mertens function? I don't see how to derive it :( – Daniel Fischer Nov 19 '13 at 11:23
  • @DanielFischer: I am quoting the quantitative version of the prime number theorem due to Hadamaard and De La Vallee Poussin. The bounds for $\pi(x)-\text{li}(x)$ and $M(x)$ will be within a logarithmic factor of each other, and so you cannot deduce the statement I wrote solely from the fact that $\pi(x)-\text{li}(x)=o\left(\frac{x}{\log x}\right)$. You need to use the stronger bound $$\pi(x)-\text{li}(x)=O\left(xe^{-c\sqrt{\log x}}\right),$$ which was originally prove in 1897. – Eric Naslund Nov 19 '13 at 13:46
  • Thanks. Now I'll go attempt to figure out the relation between $\pi(x)-\operatorname{li}(x)$ and $M(x)$. – Daniel Fischer Nov 19 '13 at 13:49
  • @DanielFischer: I strongly encourage you to do so! It is a very important connection, and it is said that phrasing things in terms of the randomness that the mobius function is the best way to understand primes. – Eric Naslund Nov 19 '13 at 15:48
  • I certainly will. If I don't figure it out by myself or with the help of what few books I have, I know whom I can ask. – Daniel Fischer Nov 19 '13 at 15:50