The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^n$.
Conclude.

Let $\psi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ such that, for all $x \in \mathbb{R}$, $$\psi(x) = \begin{cases} \dfrac{x}{\mathrm{e}^{x} - 1} & \text{if } x \neq 0 \\ 1 & \text{otherwise} \end{cases}$$ Let furthermore $u$ be the function from $\mathbb{R}^{2}$ to $\mathbb{R}$ such that, for all $(x,t) \in \mathbb{R}^{2}$, $$u(x,t) = \psi(x)\,\mathrm{e}^{tx}.$$ Show that $u$ is of class $\mathcal{C}^{\infty}$ on $\mathbb{R}^{2}$.

Let $\psi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ such that, for all $x \in \mathbb{R}$, $$\psi(x) = \begin{cases} \dfrac{x}{\mathrm{e}^{x} - 1} & \text{if } x \neq 0 \\ 1 & \text{otherwise} \end{cases}$$ Let furthermore $u$ be the function from $\mathbb{R}^{2}$ to $\mathbb{R}$ such that, for all $(x,t) \in \mathbb{R}^{2}$, $$u(x,t) = \psi(x)\,\mathrm{e}^{tx}.$$ For all $(x,t) \in \mathbb{R}^{2}$, calculate $\dfrac{\partial u}{\partial t}(x,t)$ then show that, for all $n \in \mathbb{N}^{*}$, $$\frac{\partial}{\partial t}\frac{\partial^{n} u}{\partial x^{n}}(x,t) = x\frac{\partial^{n} u}{\partial x^{n}}(x,t) + n\frac{\partial^{n-1} u}{\partial x^{n-1}}(x,t).$$

Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Let $a , b \in \mathbf { R }$ such that $a \neq 0$ and $| b | \leq | a |$. Show that
$$| a + b | = | a | + \operatorname { sign } ( a ) b$$
where $\operatorname { sign } ( x ) = x / | x |$ for nonzero real $x$. Deduce that:
$$\forall n \in \mathbf { N } ^ { * } , \quad E \left( \left| S _ { 2 n } \right| \right) = E \left( \left| S _ { 2 n - 1 } \right| \right)$$

Show that $$\forall x , y \in \mathbf { R } _ { + } , \quad x y \leq \frac { x ^ { p } } { p } + \frac { y ^ { q } } { q }$$ where $p , q \in ] 1 , + \infty [$ such that $\frac { 1 } { p } + \frac { 1 } { q } = 1$.

Show that the set of quasi-polynomial functions forms a $\mathbb{C}$-vector space.

Let $p , q \in ] 1 , + \infty [$ such that $\frac { 1 } { p } + \frac { 1 } { q } = 1$. Let $X , Y \in L ^ { 0 } ( \Omega )$ which we assume are both non-negative. Deduce the following inequality (Hölder's inequality): $$\mathbf { E } ( X Y ) \leq \left( \mathrm { E } \left( X ^ { p } \right) \right) ^ { 1 / p } \left( \mathrm { E } \left( Y ^ { q } \right) \right) ^ { 1 / q } .$$ You may begin by treating the case where $\mathbf { E } \left( X ^ { p } \right) = \mathbf { E } \left( Y ^ { q } \right) = 1$.

Show that $p \wedge p_0 = 1$ if and only if $p$ has no stable root.

Show that if $P, Q : \mathbb{Z} \rightarrow \mathbb{C}$ are two quasi-polynomial functions such that $P(n) = Q(n)$ for all $n \geq 0$, then $P = Q$.

What inequality do we recover when $p = q = 2$ ? Give a direct proof of it.

Let $Q$ be a polynomial of degree $p$. We say that $Q$ is antireciprocal if $$Q(X) = -X^p Q\left(\frac{1}{X}\right)$$ Show that if $Q$ is antireciprocal, 1 is a root of $Q$ and that there exists a polynomial $P$ that is constant or reciprocal such that $Q = (X-1)P$.

Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Justify that there exists $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.

Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Let $h$ be the polynomial of degree $n$ defined by $h(X) = X p'$, where $p'$ is the derivative polynomial of $p$. We denote by $h_0$ and $(p')_0$ the reciprocal polynomials of $h$ and $p'$ respectively.
Show that $h = np - \lambda (p')_0$, then that $h_0 = \lambda(np - Xp')$.

Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Show that: for all $t \geq 0$, for all $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { E } \left( \exp \left( t \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) \right) \leq \exp \left( \frac { t ^ { 2 } } { 2 } \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } \right)$$

Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Verify that $p'$ is split over $\mathbf{R}$ then show that $h \wedge h_0 = 1$ and deduce that $p'$ has no stable root.

Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Deduce that: for all $t \geq 0$, for all $x \geq 0$ and for all $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { P } \left( \exp \left( x \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| \right) > \mathrm { e } ^ { t x } \right) \leq 2 \mathrm { e } ^ { - t x } \exp \left( \frac { x ^ { 2 } \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } } { 2 } \right) .$$ You may use Markov's inequality.

For every integer $j \in \llbracket 1, n \rrbracket$, we denote by $f_j$ the polynomial $$f_j = a_n \prod_{k=j+1}^{n}\left(1 - \alpha_k X\right) \prod_{k=1}^{j-1}\left(X - \alpha_k\right)$$ with, according to standard conventions, $\prod_{k=n+1}^{n}(1-\alpha_k X) = \prod_{k=1}^{0}(X - \alpha_k) = 1$.
Show that if there exist two integers $i, k$ such that $1 \leq i < k \leq n$ and $\alpha_i \alpha_k = 1$, then $\alpha_i$ is a root of each polynomial $f_j$, where $j \in \llbracket 1, n \rrbracket$, and that the family $(f_1, \ldots, f_n)$ is linearly dependent.

We denote by $J_n^{(\mathrm{s})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad J_n^{(\mathrm{S})}(i,j) = \frac{2}{\sqrt{2n+1}} \sin\left(\frac{2\pi ij}{2n+1}\right).$$
Show that, for all $p \in \mathbb{N}^*$ and $x \in \mathbb{R} \backslash \pi\mathbb{Z}$, $$\sum_{k=1}^{p} \cos(2kx) = \frac{1}{2}\left(\frac{\sin((2p+1)x)}{\sin(x)} - 1\right)$$

Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Show that: for all $t \geq 0$ and for all non-zero $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { P } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| > t \right) \leq 2 \exp \left( - \frac { t ^ { 2 } } { 2 \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } } \right) .$$

Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$, we define the rational function $g_j \in E$ by $$g_j = \frac{f_j}{\prod_{i=1}^{n}(1 - \alpha_i X)}$$ and the map $P_j$, which associates to a rational function $f \in E$ the rational function $$P_j(f) = \frac{(1 - \alpha_j X)f - (1 - \alpha_j^2)f(\alpha_j)}{X - \alpha_j}$$
Show that for every $j \in \llbracket 1, n \rrbracket$, the map $P_j$ is an endomorphism of $E$ and determine its kernel.

Let $p \in \left[ 1 , + \infty \right[$. Let $X$ be a positive and finite real random variable. Let $F _ { X }$ be the function defined for all $t \geq 0$ by $$F _ { X } ( t ) = \mathbf { P } ( X > t ) .$$ Show that the integral $\int _ { 0 } ^ { + \infty } t ^ { p - 1 } F _ { X } ( t ) \mathrm { d } t$ converges, then that $$\mathbf { E } \left( X ^ { p } \right) = p \int _ { 0 } ^ { + \infty } t ^ { p - 1 } F _ { X } ( t ) \mathrm { d } t$$

Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$ and every $g \in E$, compute $P_j\left(\frac{(X - \alpha_j)g}{1 - \alpha_j X}\right)$.

Until the end of part B, we assume that no root of $p$ is stable.
Deduce that the family $(f_1, \ldots, f_n)$ is linearly independent.

Let $n \geq 1$ be an integer. A non-empty compact subset $P \subset \mathbb{R}^n$ is a polytope if there exist a non-empty finite set $I$ and if for all $i \in I$ there exist a linear form $\ell_i : \mathbb{R}^n \rightarrow \mathbb{R}$ and a real number $a_i \in \mathbb{R}$ such that $P = \{x \in \mathbb{R}^n : \ell_i(x) \leq a_i\ \forall i \in I\}$. A face $F$ of $P$ is a non-empty subset such that there exists $J \subset I$ with $F = F_J = \{x \in P : \ell_j(x) = a_j\ \forall j \in J\}$.
Verify that every face $F$ of $P$ is a polytope and that $\operatorname{dim} F < \operatorname{dim} P$ if $F \neq P$.

12. Show that, if $0 \leqslant j \leqslant n - 2$, for all $0 \leqslant i \leqslant n - j - 1$, the polynomial $p _ { i }$ is orthogonal to $\mathbb { R } _ { i - 1 } [ X ]$ for the inner product $\langle \cdot , \cdot \rangle _ { j + 1 }$.