Exercise 66, § 1

We denote by $e:\mathbf{T}\rightarrow \mathbf{C}^*$ the character defined by passing to the quotient by the character $x\mapsto e^{2i\pi x}$ of $\mathbf{R}$.

Let $d\geqslant 2$ be an integer. Let $\alpha \in \mathbf{R}$ and let $g\in \mathbf{R}[X]$ be a polynomial whose degree is $< d$. Let $a$ and $q$ be relatively prime integers, with $q\not= 0$, such that

a) Suppose that $\alpha \notin \mathbf{Z}$. Let N and M be integers with $M\geqslant 0$. We have

sin($\pi \alpha$ )

$n=N$

b) We denote $D = 2^{d-1}$. For every $\varepsilon >0$, there exists a real number $C(\varepsilon , d)\geqslant 0$ such that for every integer $P\geqslant 1$, one has

$n=1$

(Proceed by induction on $d$ by reducing the square of the left-hand term of the inequality to a family of polynomials of degree $\leqslant d-1$; when $d= 2$, apply then question a). One may prove and use the estimate

when $n\rightarrow +\infty$, where $n\geqslant 1$ and $d(n)$ is the number of divisors of $n$ in $\mathbf{N}$, valid for every real number $\varepsilon >0$.)

c) Deduce a new proof of the assertion of question c) of exercise 35.