We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Show that the set $\Lambda_X$ defined in question 8a is closed under addition and that $r\left(\Lambda_X\right) = 0$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Using the results of questions 13a and 13b, deduce that $h(-x) \rightarrow h(0)$ when $x \rightarrow +\infty$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Conclude that $h$ is a constant function.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. The function $f$ is the unique bounded and uniformly continuous solution of equation (E), and $f'$ is bounded and uniformly continuous. Prove that the function $x \mapsto \sup_{t \geqslant x} f'(t)$ admits a finite limit when $x \rightarrow +\infty$. We denote $$c := \lim_{x \rightarrow +\infty} \sup_{t \geqslant x} f'(t)$$

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. With $c := \lim_{x \rightarrow +\infty} \sup_{t \geqslant x} f'(t)$, show that there exists a sequence $y_n \rightarrow +\infty$ such that $f'\left(y_n\right) \rightarrow c$ when $n \rightarrow +\infty$.

Show that $I ( \alpha )$ is an ideal of $\mathbb { Q } [ X ]$, different from $\{ 0 \}$, where $\alpha$ is a fixed algebraic number and $I ( \alpha ) = \{ P \in \mathbb { Q } [ X ] \mid P ( \alpha ) = 0 \}$.

We fix an integer $n \geq 2$.
(a) Calculate $\Phi _ { n } ( 0 )$.
(b) Calculate $\Phi _ { n } ( 1 )$ as a function of the prime factorization of $n$. Hint: reason by induction on $n$, using question 7.

Show that $\Phi _ { n } \in \mathbb { Z } [ X ]$.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ Verify that $P _ { n }$ has no root in $\mathbb { Q }$ and that $P _ { n }$ has at least one real root strictly greater than 1. We fix such a root $\alpha _ { n }$ in the sequel.

5. In the remainder of this part, $p$ denotes a fixed odd prime integer. One may use without proof Wilson's theorem: $$(p-1)! + 1 \equiv 0 \quad [p]$$ We denote by $\mathbb{Z}_p$ the field $\mathbb{Z}/p\mathbb{Z}$ and if $a \in \mathbb{Z}$, we denote by $\bar{a}$ its class in $\mathbb{Z}_p$. For $1 \leq k \leq e_p$, we denote by $\mathcal{P}_k$ the set of subsets $P$ with $k$ elements of $\mathbb{Z}_p$ satisfying the condition $$\forall \alpha \in P, \quad \alpha + 1 \notin P.$$ a. For $P = \{\alpha_1, \ldots, \alpha_k\} \in \mathcal{P}_k$ and $\alpha \in \mathbb{Z}_p$, we set $\tau_\alpha(P) = \{\alpha_1 + \alpha, \ldots, \alpha_k + \alpha\}$. Show that the map $\alpha \mapsto \tau_\alpha$ is a homomorphism from $(\mathbb{Z}_p, +)$ to the group of bijections of $\mathcal{P}_k$. b. We define a relation $\mathscr{R}$ between elements of $\mathcal{P}_k$ as follows: if $A, B$ are in $\mathcal{P}_k$, $A \mathscr{R} B$ if and only if there exists $\alpha \in \mathbb{Z}_p$ such that $B = \tau_\alpha(A)$. Show that $\mathscr{R}$ is an equivalence relation on $\mathcal{P}_k$, and that each equivalence class has cardinality $p$ and admits a representative of the form $\{\bar{0}, \bar{a}_2, \ldots, \bar{a}_k\}$ with $0 < a_2 < \cdots < a_k < p$. We choose such a representative for each class and we denote by $R$ the set of representatives thus chosen. c. Prove that $$\overline{c_{p-1,k}} = \sum_{\{0, \ldots, a_k\} \in R} \sum_{1 \leq \ell \leq p-1} \bar{\ell}\, \overline{\ell+1}\, \overline{a_2 + \ell}\, \overline{a_2 + \ell + 1} \cdots \overline{a_k + \ell}\, \overline{a_k + \ell + 1}$$

6. a. For $q \in \mathbb{N}$, we set $S_q = \sum_{\ell=0}^{p-1} \ell^q$. Observe that $p$ divides $\sum_{\ell=0}^{p-1} ((\ell+1)^{q+1} - \ell^{q+1})$ and deduce by recursion that $p$ divides $S_q$ for $0 \leq q \leq p-2$. b. Let $Z = [z_{ij}]$ and $Z' = [z'_{ij}]$ be two square matrices of order $N$ with entries in $\mathbb{Z}$. We define the relation $Z \equiv Z'[p]$ by $z_{ij} \equiv z'_{ij}[p]$ for $1 \leq i, j \leq N$. Prove that $$(M_{p-1})^{(p-1)} \equiv (-1)^{(p-1)/2} \mathrm{Id} \quad [p].$$ c. What can be said about a polynomial $Q$ with integer coefficients such that $Q(M_{p-1}) \equiv 0[p]$?

7. We recall that $\bar{E}_n$ denotes the class of $E_n = \operatorname{Card} \operatorname{MD}(n)$ in $\mathbb{Z}_p$. a. Show that $E_{2n+1} \equiv u_{2n}[p]$, where $u_m$ is the coefficient of the term $X$ in the decomposition of $\delta_{p-1}^m(X)$ in the basis $(X, \ldots, X^{p-1})$. b. Prove that the sequence $(\bar{E}_{2n+1})_{n \in \mathbb{N}}$ is periodic, with minimal period $(p-1)/2$ if $p \equiv 1\,[4]$ and minimal period $(p-1)$ if $p \equiv 3\,[4]$. c. Indicate the modifications to be made to the preceding questions to show an analogous result for the sequence $(\bar{E}_{2n})_{n \in \mathbb{N}}$.

Justify that, for all $n \in \mathbb{N}^{*}$,
$$( f * g ) ( n ) = \sum _ { \left( d _ { 1 } , d _ { 2 } \right) \in \mathcal { C } _ { n } } f \left( d _ { 1 } \right) g \left( d _ { 2 } \right)$$
where $\mathcal{C}_n = \left\{ \left( d_1, d_2 \right) \in \left( \mathbb{N}^* \right)^2 \mid d_1 d_2 = n \right\}$.

The purpose of this question is to show that $\sqrt { 3 }$ is not an eigenvalue of a matrix in $S _ { 2 } ( \mathbb { Q } )$. We assume that there exists $M \in S _ { 2 } ( \mathbb { Q } )$ such that $\sqrt { 3 }$ is an eigenvalue of $M$.
2a. Using the irrationality of $\sqrt { 3 }$, show that the characteristic polynomial of $M$ is $X ^ { 2 } - 3$.
2b. Show that if $n \in \mathbb { Z }$, then $n ^ { 2 }$ is congruent to 0 or 1 modulo 3.
2c. Show that there does not exist a triple of integers $( x , y , z )$ that are coprime as a whole such that $x ^ { 2 } + y ^ { 2 } = 3 z ^ { 2 }$.
2d. Conclude.

Similarly, by exploiting the set $\mathcal{C}_n^{\prime} = \left\{ \left( d_1, d_2, d_3 \right) \in \left( \mathbb{N}^* \right)^3 \mid d_1 d_2 d_3 = n \right\}$, show that $*$ is associative.

The purpose of this question is to show that $\sqrt [ 3 ] { 2 }$ is not an eigenvalue of a symmetric matrix with coefficients in $\mathbb { Q }$. We reason by contradiction, assuming the existence of a matrix $M \in S _ { n } ( \mathbb { Q } )$ (for some integer $n$) for which $\sqrt [ 3 ] { 2 }$ is an eigenvalue.
4a. Show that $X ^ { 3 } - 2$ divides the characteristic polynomial of $M$. (One may begin by proving that $\sqrt [ 3 ] { 2 } \notin \mathbb { Q }$.)
4b. Conclude.

Let $f$ and $g$ be two multiplicative functions. Show that if
$$\forall p \in \mathcal{P}, \quad \forall k \in \mathbb{N}^*, \quad f\left(p^k\right) = g\left(p^k\right)$$
then $f = g$.

Let $m$ and $n$ be two non-zero natural integers that are coprime. Show that the map
$$\pi : \left\lvert\, \begin{gathered} \mathcal{D}_n \times \mathcal{D}_m \rightarrow \mathcal{D}_{mn} \\ \left( d_1, d_2 \right) \mapsto d_1 d_2 \end{gathered} \right.$$
is well-defined and realizes a bijection between $\mathcal{D}_n \times \mathcal{D}_m$ and $\mathcal{D}_{mn}$.

Deduce that if $f$ and $g$ are two multiplicative functions, then $f * g$ is still multiplicative.

Let $f$ be a multiplicative function. Show that there exists a multiplicative function $g$ such that, for all $p \in \mathcal{P}$ and all $k \in \mathbb{N}^*$,
$$g\left(p^k\right) = -\sum_{i=1}^{k} f\left(p^i\right) g\left(p^{k-i}\right)$$
and that it satisfies $f * g = \delta$.

Let $\mu$ be the arithmetic function defined by
$$\mu(n) = \begin{cases} 1 & \text{if } n = 1 \\ (-1)^r & \text{if } n \text{ is the product of } r \text{ distinct prime numbers} \\ 0 & \text{otherwise} \end{cases}$$
Show that $\mu$ is multiplicative.

Let $\mu$ be the Möbius function defined by
$$\mu(n) = \begin{cases} 1 & \text{if } n = 1 \\ (-1)^r & \text{if } n \text{ is the product of } r \text{ distinct prime numbers} \\ 0 & \text{otherwise} \end{cases}$$
Show that $\mu * \mathbf{1} = \delta$.

Let $f \in \mathbb{A}$, and let $F \in \mathbb{A}$ such that, for all $n \in \mathbb{N}^*, F(n) = \sum_{d \mid n} f(d)$. Show that, for all $n \in \mathbb{N}^*$,
$$f(n) = \sum_{d \mid n} \mu(d) F\left(\frac{n}{d}\right)$$

We denote $\varphi$ the Euler totient function, defined by:
$$\forall n \in \mathbb{N}^*, \quad \varphi(n) = \operatorname{card}\{k \in \llbracket 1, n \rrbracket \mid k \wedge n = 1\}$$
Prove that $\varphi = \mu * \mathbf{I}$.

Let $f$ be an arithmetic function, $n \in \mathbb{N}^*$ and $g = f * \mu$. We denote $M = \left(m_{ij}\right)$ the matrix of $\mathcal{M}_n(\mathbb{C})$ with general term $m_{ij} = f(i \wedge j)$. We also define the divisor matrix $D = \left(d_{ij}\right)$ by:
$$d_{ij} = \begin{cases} 1 & \text{if } j \text{ divides } i \\ 0 & \text{otherwise} \end{cases}$$
Let $M'$ be the matrix with general term $m_{ij}' = \begin{cases} g(j) & \text{if } j \text{ divides } i, \\ 0 & \text{otherwise.} \end{cases}$
Show that $M = M' D^\top$, where $D^\top$ is the transpose of $D$.