We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$, with $g = f^\dagger$ the reciprocal series. Show that $\hat{g} \prec (1/\lambda)(I + \hat{F} \circ \hat{g})$, conclude using part C that $\rho(g) > 0$ if $\rho(f) > 0$.

The set of real symplectic matrices is defined as $$\mathrm { Sp } _ { n } ( \mathbb { R } ) = \mathrm { Sp } _ { 2 m } ( \mathbb { R } ) = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid M ^ { \top } J M = J \right\}$$ where $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $\mathrm { Sp } _ { n } ( \mathbb { R } )$ is a subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$, stable under transposition and containing the matrix $J$.

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $\forall z \in \mathbb{U},\ Q(z) \neq 0$. For $t \in [-\pi, \pi]$, we set $f(t) = F(e^{it}) = g(t) + ih(t)$. For $u \in [-\pi, \pi]$, we define $f_{u}(t) = g(t)\cos(u) + h(t)\sin(u)$.
Express the integral $$\int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}u\right) \mathrm{d}t$$ in terms of $V(f)$.

Let $(\Omega, \mathscr{A}, P)$ be a probability space. If $N \in \mathbb{N}^*$ and $p$ is a prime number, we denote by $\nu_p(N)$ the $p$-adic valuation of $N$. For $n \in \mathbb{N}^*$, we define the map $$\psi_n : \mathbb{N}^* \longrightarrow \mathbb{N}^*, \quad x \longmapsto \prod_{i=1}^{n} p_i^{\nu_{p_i}(x)}$$ where $(p_i)_{i \in \mathbb{N}^*}$ is the sequence of prime numbers, ordered in increasing order.
Let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Show that $$\forall x \in \mathbb{N}^*, \quad P(X = x) = \lim_{n \rightarrow +\infty} P(\psi_n(X) = x).$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $p \in \mathbb{N}^*$ and $M \in \mathcal{M}_{p,n}(\mathbb{R})$. We define the discrete random variable $Z = MY$, with values in $\mathcal{M}_{p,1}(\mathbb{R})$. Justify that $Z$ admits an expectation and express $\mathbb{E}(Z)$ in terms of $\mathbb{E}(Y)$. Show that $Z$ admits a covariance matrix $\Sigma_Z$ and that $$\Sigma_Z = M \Sigma_Y M^\top.$$

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality.
Let $n \in \mathbf { N }$. Show that $P _ { n , N }$ is included in $\llbracket 0 , n \rrbracket ^ { N }$ and non-empty for all $N \in \mathbf { N } ^ { * }$, that the sequence $\left( p _ { n , N } \right) _ { N \geq 1 }$ is increasing and that it is constant from rank $\max ( n , 1 )$ onwards.

We consider the special case of a series of the form $f = I + F, F \in O_2$. We denote $G := f^\dagger - I, G \in O_2$. Show that $[G]_{d+1} + F \circ (I + [G]_d) \in O_{d+2}$ for all $d \geqslant 1$ (the notation $[f]_d$ is defined in the introduction to the subject).

Let $A , B , C , D$ be in $\mathcal { M } _ { m } ( \mathbb { R } )$ and let $M = \left( \begin{array} { l l } A & B \\ C & D \end{array} \right) \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ (block decomposition), where $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $M \in \mathrm { Sp } _ { 2 m } ( \mathbb { R } )$ if and only if
$$A ^ { \top } C \text { and } B ^ { \top } D \text { are symmetric } \quad \text { and } \quad A ^ { \top } D - C ^ { \top } B = I _ { m } .$$

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $\forall z \in \mathbb{U},\ Q(z) \neq 0$. For $t \in [-\pi, \pi]$, we set $f(t) = F(e^{it}) = g(t) + ih(t)$. For $u \in [-\pi, \pi]$, we define $f_{u}(t) = g(t)\cos(u) + h(t)\sin(u)$.
We admit the equality $$\int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}u\right) \mathrm{d}t = \int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}t\right) \mathrm{d}u$$ We also admit that, for $u \in [-\pi, \pi]$ such that $f_{u}$ is not constant, the set of points in $]-\pi, \pi[$ where the function $f_{u}^{\prime}$ vanishes is finite.
Deduce the inequality $$(3) \quad V(f) \leq 2\pi n \|f\|_{\infty}.$$

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We assume that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$. We want to show that $\mu_1 = \mu_2$.
We recall that we denote by $(p_i)_{i \in \mathbb{N}^*}$ the sequence of prime numbers, ordered in increasing order. Show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\bigcup_{i=1}^{n+1} \mathbb{N}^* r p_i = \left(\bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) \cup \left(\mathbb{N}^* r p_{n+1} \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_{n+1} p_i\right).$$

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We assume that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$.
We recall that we denote by $(p_i)_{i \in \mathbb{N}^*}$ the sequence of prime numbers, ordered in increasing order.
Show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\mu_1\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) = \mu_2\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right).$$

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We assume that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$.
Using the results of 17a and 17b, conclude that $\mu_1 = \mu_2$.

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. Using the results of 17a and 17b, conclude that $\mu_1 = \mu_2$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. We denote by $P$ the change of basis matrix from the canonical basis of $\mathcal{M}_{n,1}(\mathbb{R})$ to an orthonormal basis formed by eigenvectors of $\Sigma_Y$. We define the discrete random variable $X = P^\top Y = \left(\begin{array}{c} X_1 \\ \vdots \\ X_n \end{array}\right)$.
Prove that $\Sigma_X$ is a diagonal matrix.

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. We denote by $p _ { n }$ the final value of $\left( p _ { n , N } \right) _ { N \geq 1 }$.
Let $N \in \mathbf { N } ^ { * }$. Give a sequence $\left( a _ { n , N } \right) _ { n \in \mathbf { N } }$ such that
$$\forall z \in D , \frac { 1 } { 1 - z ^ { N } } = \sum _ { n = 0 } ^ { + \infty } a _ { n , N } z ^ { n }$$
Deduce, by induction, the formula
$$\forall N \in \mathbf { N } ^ { * } , \forall z \in D , \prod _ { k = 1 } ^ { N } \frac { 1 } { 1 - z ^ { k } } = \sum _ { n = 0 } ^ { + \infty } p _ { n , N } z ^ { n }$$

We consider the special case of a series of the form $f = I + F, F \in O_2$. We denote $G := f^\dagger - I, G \in O_2$. Assume that there exist $s > 0$ and $\alpha \in ]0,1[$ such that $\hat{F}(s) \leqslant \alpha s$. Show then that for all $d \geqslant 2, \widehat{[G]_d}((1-\alpha)s) \leqslant \alpha s$. Conclude that $$\hat{G}((1-\alpha)s) \leqslant \alpha s.$$

Show that $\mathrm { Sp } _ { 2 } ( \mathbb { R } ) = \mathrm { SL } _ { 2 } ( \mathbb { R } )$.

Let $M \in \mathcal{B}_{n}$. We define $$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$ where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$, and $b^{\prime}(M) \leq b(M)$.
We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$.
Show that there exists an element $F_{r}$ of $\mathcal{R}_{n}$ whose poles are all in $\mathbb{D}_{1/r}$ and such that the following two properties are satisfied: $$\begin{gathered} \forall z \in \mathbf{C} \backslash \mathbb{D}_{1/r}, \quad \left|F_{r}(z)\right| \leq \frac{b^{\prime}(M)}{r|z|-1} \\ \forall k \in \mathbf{N}, \quad X^{T}M^{k}Y = \frac{r^{k+1}}{2\pi} \int_{-\pi}^{\pi} F_{r}\left(e^{it}\right) e^{i(k+1)t} \mathrm{~d}t \end{gathered}$$

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$ and let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Assume that:

1. [i.] The sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ is tight.
2. [ii.] For all $r \in \mathbb{N}^*$, $\lim_{n \rightarrow +\infty} \mathbf{P}(r \mid X_n) = \mathbf{P}(r \mid X)$.

Show that the sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ converges to $\mu_X$ in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$.

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. We denote by $p _ { n }$ the final value of $\left( p _ { n , N } \right) _ { N \geq 1 }$, and $P ( z ) := \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$ for all $z \in D$.
We fix $\ell \in \mathbf { N }$ and $x \in [ 0,1 [$. Using the result of the previous question, establish the bound $\sum _ { n = 0 } ^ { \ell } p _ { n } x ^ { n } \leq P ( x )$. Deduce the radius of convergence of the power series $\sum _ { n } p _ { n } z ^ { n }$.

We set $\lambda = (f)_1$ and denote $f = \lambda z + F$, with $F \in O_2$. We assume that $\lambda \neq 0$ and that $\lambda$ is not a complex root of unity, that is, $\lambda^n \neq 1$ for all integer $n \geqslant 1$. We propose to show that there exists a unique power series of the form $h = I + H, H \in O_2$ satisfying $h^\dagger \circ f \circ h = \lambda z$. Show that there exists a unique series $H \in O_2$ such that $H \circ (\lambda I) - \lambda H = F \circ (I + H)$.

Let $M \in \mathcal{B}_{n}$. We define $$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$ where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$.
We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$.
By using the previous question, integration by parts and inequality (3) from question $17$, show that $$\forall k \in \mathbf{N}, \quad \left|X^{T}M^{k}Y\right| \leq \frac{r^{k+1}}{(k+1)(r-1)} n b^{\prime}(M).$$

Let $s \in \mathbb{N}^*$. For $n \in \mathbb{N}$, let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. We denote $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ the gcd of $X_n^{(1)}, \ldots, X_n^{(s)}$. For $r \in \mathbb{N}^*$ and $i \in \{1, 2, \ldots, s\}$, calculate $\mathbf{P}(r \mid X_n^{(i)})$ and show that $\mathbf{P}(r \mid X_n^{(i)}) \leqslant \frac{1}{r}$. Deduce that $$\lim_{n \rightarrow +\infty} \mathbf{P}\left(r \mid Z_n^{(s)}\right) = \frac{1}{r^s}$$

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. We denote by $p _ { n }$ the final value of $\left( p _ { n , N } \right) _ { N \geq 1 }$, and $P ( z ) := \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$ for all $z \in D$.
Let $z \in D$. By examining the difference $\sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n } - \sum _ { n = 0 } ^ { + \infty } p _ { n , N } z ^ { n }$, prove that
$$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$

We set $\lambda = (f)_1$ and denote $f = \lambda z + F$, with $F \in O_2$. We assume that $\lambda \neq 0$ and that $\lambda$ is not a complex root of unity. We have shown in question (19) that there exists a unique series $H \in O_2$ such that $H \circ (\lambda I) - \lambda H = F \circ (I + H)$. Conclude that there exists a unique power series of the form $h = I + H, H \in O_2$ satisfying $h^\dagger \circ f \circ h = \lambda z$.