Exercise 10, § 3

Define the von Mangoldt function on $\mathbf{N}^*$ by $\Lambda (n) = 0$ if $n$ is not of the form $p^k$ where $p$ is prime and $k\geqslant 1$, and $\Lambda (p^k) =$ log($p$).

Denote by $\zeta$ the holomorphic function on $\mathbf{C}-\{1\}$ whose restriction to the set of the $s\in \mathbf{C}$ such that $\mathscr{R}(s)>1$ coincides with the Riemann zeta function (exercise 16 of II, p. 266).

For $x >0$ put

a) Now prove that for all $s\in \mathbf{C}$ such that $\mathscr{R}(s)>1$, one has

$n\geqslant 1$ b) One has

lim$_{x\rightarrow+\infty}\frac{\psi(x)}{x}= 1$.

(Use exercise 16 of II, p. 266 and exercise 9).

c) Let $\pi (x)$ be the number of prime numbers $\leqslant x$. Now prove that

$\pi (x)\sim$ log($xx$)

when $x\rightarrow +\infty$ ("prime number theorem", due to Hadamard and de la Vallée Poussin).