We assume that $\lambda$ is a real distinct from 1 and we set $w = \frac{1}{\lambda - 1}$. We further set $\mathbf{f} = (1 + w)\delta - w\mathbf{1}$.
Using the notations of Dirichlet series given in subsection I.E, express, for values of the real $s$ to be specified, $L_{\mathbf{f}}(s)$ in terms of $w$ and $L_{\mathbf{1}}(s)$.

We assume that $\lambda$ is a real distinct from 1 and we set $w = \frac{1}{\lambda - 1}$. We denote $\log_2$ the logarithm function in base 2. For $n \geq 1$, we set $S_k(n) = \sum_{m=2}^{n} D_k(m)$. Deduce from the previous question that
$$\chi_n(\lambda) = (\lambda - 1)^n - \sum_{k=1}^{\lfloor \log_2 n \rfloor} (\lambda - 1)^{n-k-1} S_k(n).$$

Assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Show that $f * g$ is defined on $\mathbb{R}$ and that $$\forall x \in \mathbb{R}, \quad (f*g)(x) = \int_{-\infty}^{+\infty} f(x-t)g(t)\,\mathrm{d}t = (g*f)(x)$$

The Pochhammer symbol is defined, for any real number $a$ and any natural integer $n$, by $$[a]_n = \begin{cases} 1 & \text{if } n = 0 \\ a(a+1)\cdots(a+n-1) = \prod_{k=0}^{n-1}(a+k) & \text{otherwise} \end{cases}$$ Let $n \in \mathbb{N}$. Give an expression of $[a]_n$

• using factorials when $a \in \mathbb{N}^*$;
• using two values of the function $\Gamma$, when $a \in D$.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Express the function $x \mapsto F_{\frac{1}{2}, 1, \frac{3}{2}}\left(-x^2\right)$ using usual functions.

Deduce that, for every polynomial $P \in \mathbb{R}[X]$, $$\lim_{n \rightarrow +\infty} \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} P\left(\Lambda_{i,n}\right)\right) = \frac{1}{2\pi} \int_{-2}^{2} P(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$

Give without proof the value of $C _ { 3 }$ and represent all Dyck paths of length 6.

For all $n \in \mathbb { N }$, set $\mu _ { n } = \left( X ^ { n } \mid 1 \right)$ with the inner product $$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$ Deduce that $\forall n \in \mathbb { N } , \mu _ { n } = C _ { n }$.

Let $n \in \mathbb { N }$. Deduce from the previous parts the value of the determinant $$H _ { n } = \operatorname { det } \left( C _ { i + j - 2 } \right) _ { 1 \leqslant i , j \leqslant n + 1 } = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \ddots & \therefore & \vdots \\ \vdots & \therefore & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & \therefore & \ddots & & \therefore & C _ { 2 n - 1 } \\ C _ { n } & C _ { n + 1 } & \cdots & C _ { 2 n - 2 } & C _ { 2 n - 1 } & C _ { 2 n } \end{array} \right|.$$

Using the relation $b_0 = 1$ and $\sum_{p=0}^{n-1} \binom{n}{p} b_p = 0$ for all integer $n \geqslant 2$, deduce the value of $b_1, b_2, b_3$ and $b_4$.

The polynomials $B_m$ are defined by $$\forall m \in \mathbb{N}, \quad B_m(x) = \sum_{k=0}^m \binom{m}{k} b_k x^{m-k}.$$
Determine $B_0, B_1, B_2$ and $B_3$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Deduce that $$\zeta(s)^{-1} = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n}\left(1 - p_k^{-s}\right).$$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
We denote $r(n)$ the number of divisors $d \geqslant 1$ of $n$. Show that the series $\sum_{n=1}^{+\infty} r(n) n^{-s}$ converges and that its sum equals $\zeta(s)^2$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$. We recall that $\chi_4(2n) = 0$ and $\chi_4(2n-1) = (-1)^{n-1}$ for $n \in \mathbb{N}^*$. We set $g(n) = r_1(n) - r_3(n)$ where $r_i(n) = \operatorname{Card}\{d \in \mathbb{N} : d \equiv i [4] \text{ and } d \mid n\}$.
Deduce that the series $$\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^s}$$ is convergent and that its sum equals $E(g(X))$.

Deduce that, for all $x \in \mathbb{R}$, $$\sin(\pi x) = \pi x \lim_{n \rightarrow +\infty} \prod_{k=1}^{n}\left(1 - \frac{x^2}{k^2}\right).$$

Show that for all $x \in ]0,1[$: $$\frac{\pi}{\sin(\pi x)} = \sum_{n=0}^{+\infty} \frac{(-1)^n}{n+x} + \sum_{n=0}^{+\infty} \frac{(-1)^n}{n+1-x}.$$

We set $C = \exp\left(\frac{I}{2\pi}\right)$ with: $$I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} + 1\right|\right\}\right) d\theta$$ Show that: $$I = 4\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}$$ You may use the result from question 2.13.

We fix a real $\alpha > 0$ and an integer $n \geq 1$. Subject to existence, we set
$$S _ { n , \alpha } ( t ) : = \sum _ { k = 1 } ^ { + \infty } \frac { k ^ { n } e ^ { - k t \alpha } } { \left( 1 - e ^ { - k t } \right) ^ { n } }$$
We also introduce the function
$$\varphi _ { n , \alpha } : x \in \mathbf { R } _ { + } ^ { * } \mapsto \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } }$$
Show, for all real $t > 0$, the existence of $S _ { n , \alpha } ( t )$, its strict positivity, and the identity
$$\int _ { 0 } ^ { + \infty } \varphi _ { n , \alpha } ( x ) \mathrm { d } x = t ^ { n + 1 } S _ { n , \alpha } ( t ) - \sum _ { k = 0 } ^ { + \infty } \int _ { k t } ^ { ( k + 1 ) t } ( x - k t ) \varphi _ { n , \alpha } ^ { \prime } ( x ) \mathrm { d } x$$
Deduce that
$$S _ { n , \alpha } ( t ) = \frac { 1 } { t ^ { n + 1 } } \int _ { 0 } ^ { + \infty } \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } } \mathrm {~d} x + O \left( \frac { 1 } { t ^ { n } } \right) \quad \text { as } t \rightarrow 0 ^ { + }$$

Using a series expansion under the integral, show that
$$\int _ { 0 } ^ { + \infty } \ln \left( 1 - e ^ { - u } \right) \mathrm { d } u = - \frac { \pi ^ { 2 } } { 6 }$$

We have $P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$ for all $z \in D$, where $p_n$ denotes the number of partitions of $n$.
Let $n \in \mathbf { N }$. Show that for all real $t > 0$,
$$p _ { n } = \frac { e ^ { n t } P \left( e ^ { - t } \right) } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta \tag{1}$$

Using a series expansion under the integral, show that $$\int_{0}^{+\infty} \ln(1-e^{-u}) \mathrm{d}u = -\frac{\pi^2}{6}$$

Let $n \in \mathbf{N}$. Show that for all real $t > 0$, $$p_n = \frac{e^{nt} P(e^{-t})}{2\pi} \int_{-\pi}^{\pi} e^{-in\theta} \frac{P(e^{-t}e^{i\theta})}{P(e^{-t})} \mathrm{d}\theta \tag{1}$$

Explicitly determine $F^{\prime\prime}(x)$, where $F(x) = \sum_{n=1}^{+\infty} f_n(x)$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$.

Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$.
Let $M \in \mathcal{B}_{n}$, $r \in ]1, +\infty[$ and $(X,Y) \in \mathcal{M}_{n,1}(\mathbf{C})^{2}$. Determine a sequence of complex numbers $(c_{j})_{j \in \mathbf{N}}$ such that the series $\sum c_{j}$ converges absolutely and that $$\forall t \in \mathbf{R}, \quad X^{T}R_{re^{it}}(M)Y = \sum_{j=0}^{+\infty} c_{j} e^{-i(j+1)t}.$$ If $k \in \mathbf{N}$, deduce, using question 9, an integral expression for $X^{T}M^{k}Y$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)! \zeta(2n)}{2^{2n-1}\pi^{2n}}.$$
Calculate $b_2$, $b_4$ and $b_6$ then $\zeta(2)$, $\zeta(4)$ and $\zeta(6)$.