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(a) Show that $\int_2^x\frac{\pi(t)}{t^2}dt=\sum_{p\leq x }\frac{1}{p}+o(1)\sim\log\log x.$

(b) Let $\rho(x)$ be the ratio of the two functions involved in the prime number theorem:

$$\rho(x)=\frac{\pi(x)}{x/ \log x}$$ Show that for no $\delta>0$ is there a $T=T(\delta)$ such that $\rho(x)>1+\delta$ for all $x>T$, nor is there a $T$ such that $\rho(x)<1-\delta$ for all $x>T$. This means that $$\lim \inf \rho(x)\leq 1 \leq \lim \sup \rho(x),$$ so that if $\lim \rho(x)$ exists, it must have the value $1$.

I don't know how to apply (a) to (b), and I couldn't find any sources related to such a proof. Could you give me a proof on that?

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    Are you able to show part (a)?2011-12-14
  • 0
    @Srivatsan Yes, I could.2011-12-14

1 Answers 1