Exercise 11, § 3

Define the Möbius function $\mu$ on $\mathbf{N}^*$ by $\mu(n) = (-1)^k$ if $n$ is the product of $k$ pairwise distinct prime factors and $\mu(n) = 0$ otherwise.

For $x >0$ put

$n\leqslant x$

a) Now prove that for $\mathscr{R}(s)>1$, one has

$n\geqslant 1$

b) Now prove that $f$ is bounded, and that $f(y)-f(x) = O((y-x)/x)$ for $y\geqslant x\geqslant 1$, so that $f$ is slowly oscillating on the group $\mathbf{R}^*_+$ when $x$ tends to $+\infty$.

c) Let $a, b\in \mathbf{R}_+^*$. For $x >0$, let $[x]$ be the integer part of $x$. Put $g_0(t) = [t^{-1}]$ for $t\in \mathbf{R}_+^*$ and

The function $g$ is integrable on $\mathbf{R}^*_+$ for Lebesgue measure. If $s\in \mathbf{C}$ and $\mathscr{R}(s)>0$, one has

d) One has $\int_0^{+\infty}g(t)t^{ix}dt\not= 0$ for all $x\in \mathbf{R}$. (Use part j) of exercise 16 of II, p. 266.)

e) Now prove that $\int_0^{+\infty}g(t/x)f(t)dt=o(x)$ when $x\rightarrow +\infty$. Deduce that $M(x) =o(x)$ when $x\rightarrow +\infty$.

(One can show in an elementary manner that this result is equivalent to the prime number theorem, cf. exercise 10; see for example H. Iwaniec and E. Kowalski, Analytic number theory, A.M.S Coll. Publ. 53 (2004), p. 31– 32.)