Problem from Apostol's analytic number theory book

Im trying to solve the exercise 13.2 in Apostol's analytic number theory book: Let $A(x)=\sum_{n\leq x}a(n)$, where $a(n)$ is zero unless $n=p^k$ for some prime $p$, in that case $a(n)=1/k$. Prove that $A(x)=\pi(x)+O(\sqrt{x}\log \log x)$.

I've tried things like this: By the definition of the function $a$, we have that \begin{equation*}A(x)=\underset{p\leq x}{\sum}\sum_{\substack{k\in \mathbb{Z}^+\\ p^k\leq x}}a(p^k)+1=\underset{p\leq x}{\sum}\sum_{\substack{k\in \mathbb{Z}^+\\ p^k\leq x}}\frac{1}{k}.\end{equation*} The last formula holds for $x\geq 1$. Look that, for all $k\in \mathbb{Z}^+$, we have the following $$p^k\leq x \iff k\log p=\log (p^k)\leq \log x \iff k \leq \log_p(x).$$ Here $\log_p x:=\frac{\log x}{\log p}$. By all this and theorem 3.2 from Apostol, we get

\begin{eqnarray*} A(x) & = & \underset{p\leq x}{\sum}\sum_{\substack{k\in \mathbb{Z}^+\\ p^k\leq x}}\frac{1}{k}= \underset{p\leq x}{\sum}\sum_{\substack{k\in \mathbb{Z}^+ \\ k\leq \log _p x}}\frac{1}{k}= \underset{p\leq x}{\sum}[\log \log_p x + C + O(\frac{\log p}{\log x})]\\ & = & \underset{p\leq x}{\sum}O(\log \log x)+\underset{p\leq x}{\sum}C+\underset{p\leq x}{\sum}O(\log^{-1}x)\\ & = & O(\log \log x)\underset{p\leq x}{\sum}1+C\underset{p\leq x}{\sum}1+O(\log^{-1}x)\underset{p\leq x}{\sum}1\\ & = & O(\log \log x)\pi(x)+(C-1+1)\pi(x)+O(\log^{-1}x)\pi(x)\\ & = & O(\log \log x)O(\frac{x}{\log x})+(C-1)O(\frac{x}{\log x})+\pi(x)+O(\log^{-1}x)O(\frac{x}{\log x}). \end{eqnarray*}

First, look that if $k$ is a positive integer suchthat $p^k \leq x$ for some prime $p$, then $2^k \leq p^k \leq x$, so $k\log 2=\log 2^k\leq \log x$ and $k\leq \log_2 x$.

Therefore

\begin{eqnarray*} A(x) & = & \sum_{k\in \mathbb{Z}^{+}} \sum_{\substack{p \text{ prime}\\ p^k \leq x}} \frac{1}{k}=\sum_{k=1}^{[\log_2 x]}\frac{1}{k}\sum_{\substack{p \text{ prime}\\ p \leq x^{1/k}}}1\\ & = & \sum_{k=1}^{[\log_2 x]}\frac{\pi(x^{1/k})}{k}\\ & = & \pi(x)+\sum_{k=2}^{[\log_2 x]}\frac{\pi(x^{1/k})}{k}\\ & = & \pi(x) +\sum_{k=2}^{[\log_2 x]} \frac{\overbrace{x^{1/k}/\log x^{1/k}+o(x^{1/k}/\log x^{1/k})}^{\text{by the prime number theorem}}}{k}\\ & = & \pi(x)+\sum_{k=2}^{[\log_2 x]}\frac{x^{1/k}}{k \log x^{1/k}}+ \sum_{k=2}^{[\log_2 x]}\frac{\overbrace{O(x^{1/k}/\log x^{1/k})}^{\text{every }o\text{ is }O}}{k}\\ & = & \pi(x)+\frac{1}{\log x}\sum_{k=2}^{[\log_2 x]} x^{1/k}+O(\sum_{k=2}^{[\log_2 x]}\frac{x^{1/k}/\log x^{1/k}}{k}) \\ &=& \pi(x)+\frac{1}{\log x}O(\sqrt{x}\log x)+O(\frac{1}{\log x}\sum_{k=2}^{[\log_2 x]} x^{1/k})\\ &=&\pi(x) +O(\sqrt{x})+O(\frac{1}{\log x}O(\sqrt{x}\log x))\\ &=& \pi(x) +O(\sqrt{x})=\pi(x) +O(\sqrt{x}\log \log x). \end{eqnarray*}

In the seventh equality we used the following $$\sum_{k=2}^{[\log_2 x]}x^{1/k}\leq \sum_{k=2}^{[\log_2 x]}\sqrt{x}\leq \sqrt{x}\log_2 x=\frac{\sqrt{x}\log x}{\log 2} \text{ for all }x\geq 1.$$

• That is a very well formatted question, which is rare these days. Congratulations. – Klangen Jul 25 '17 at 21:22

Unfortunately, once you have error terms like $O(x/\log x)$, you're not going to be able to recover and obtain the desired $O(\sqrt x \log\log x)$.
I suggest that you first establish the identity $$A(x) = \pi(x) + \tfrac12\pi(\sqrt x) + \tfrac13\pi(\sqrt[3]x) + \cdots.$$ (Hint: instead of sorting first by $p$, sort first by $k$.) From there you should be able to obtain the desired result (indeed, if you're careful, you can get $O(\sqrt x/\log x)$ even).
PS: I think $a(1)=0$.