Show that, for all $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$, the series $\sum _ { n \geq 1 } \frac { e ^ { \mathrm { i } n \theta } } { n }$ converges.

Justify that the series with general term $a _ { n } = \frac { 1 } { n } - \int _ { n - 1 } ^ { n } \frac { \mathrm {~d} t } { t }$ converges.

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k } = 1 + \frac { 1 } { 2 } + \cdots + \frac { 1 } { n }$. We denote $\zeta$ the function defined for $x > 1$ by $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$.
Let $r$ be a natural integer. For which values of $r$ is the series $\sum _ { n \geqslant 1 } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ convergent?
In the rest of the problem we will denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Specify the domain of convergence of the series $\sum _ { k \geqslant 1 } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k }$ and show that the sequence $\left( R _ { N } ( x ) \right) _ { N \geqslant 1 }$ is not bounded.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, and $$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$ We admit that $\Gamma(x) \underset{x \rightarrow +\infty}{\sim} \sqrt{2\pi} x^{(x-1/2)} \mathrm{e}^{-x}$.
Investigate whether the series on the right-hand side of $(S)$ converges absolutely when $|x| = R$, where $R$ is the radius of convergence.

Let $\left(u_{n}\right)_{n \in \mathbb{N}}$ be an element of $E$ whose convergence rate is of order $r$, where $r$ is a real strictly greater than 1. Show that the convergence of the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is fast.

a) Show that the sequence $\left(S_{n}\right)_{n \in \mathbb{N}}$ defined by $\forall n \in \mathbb{N}, S_{n}=\sum_{k=0}^{n} \frac{1}{k!}$ is an element of $E$. We denote by $s$ the limit of this sequence.
b) Show that for every natural integer $n$, we have $\frac{1}{(n+1)!} \leqslant s-S_{n} \leqslant \frac{1}{(n+1)!} \sum_{k=0}^{+\infty} \frac{1}{2^{k}}$.
c) Deduce that the convergence of the sequence $\left(S_{n}\right)_{n \in \mathbb{N}}$ is fast.
d) Let $r$ be a real strictly greater than 1. Show that the convergence of the sequence $\left(S_{n}\right)_{n \in \mathbb{N}}$ towards $s$ is not of order $r$.

Determine $\mathcal{D}_{\zeta}$, the domain of definition of $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Determine $\mathcal{D}_{f}$, the domain of definition of $f$.

For every natural integer $n$ and every real $x$ in $J = [0, 1/2[$, set $$S_n(x) = \sum_{p=1}^{+\infty} \left(\sum_{k=n+1}^{+\infty} \frac{2^{2p+1} x^{2p-1}}{(2k-1)^{2p}}\right).$$ Justify that, for every natural integer $n$, the function $S_n$ is defined on $J$.

Let $f$ be an arithmetic function. We define, for all real $s$ such that the series converges,
$$L_f(s) = \sum_{k=1}^{\infty} \frac{f(k)}{k^s}$$
We call abscissa of convergence of $L_f$
$$A_c(f) = \inf\left\{s \in \mathbb{R} \mid \text{the series } \sum \frac{f(k)}{k^s} \text{ converges absolutely}\right\}.$$
Show that if $s > A_c(f)$, then the series $\sum \frac{f(k)}{k^s}$ converges absolutely.

Let $f$ be an arithmetic function. We define, for all real $s$ such that the series converges,
$$L_f(s) = \sum_{k=1}^{\infty} \frac{f(k)}{k^s}$$
The abscissa of convergence of $L_f$ is defined as
$$A_c(f) = \inf\left\{s \in \mathbb{R} \mid \text{the series } \sum \frac{f(k)}{k^s} \text{ converges absolutely}\right\}.$$
Show that if $s > A_c(f)$, then the series $\sum \frac{f(k)}{k^s}$ converges absolutely.

Using the random variable $T$, show that the series $\sum _ { n \geqslant 0 } \frac { C _ { n } } { 4 ^ { n } }$ converges.

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 $s$. If $n \in \mathbb{N}^*$, 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 $\sum_{n=1}^{+\infty} g(n) n^{-s}$ converges.

Show that $| L ( z ) | \leq - \ln ( 1 - | z | )$ for all $z$ in $D$. Deduce the convergence of the series $\sum _ { n \geq 1 } L \left( z ^ { n } \right)$ for all $z$ in $D$. In what follows, we denote, for $z$ in $D$,
$$P ( z ) : = \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$$

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
$$\varphi _ { n , \alpha } : x \in \mathbf { R } _ { + } ^ { * } \mapsto \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } }$$
which is obviously of class $\mathcal { C } ^ { \infty }$.
Show that $\varphi _ { n , \alpha }$ and $\varphi _ { n , \alpha } ^ { \prime }$ are integrable on $] 0 , + \infty [$.

Show that $| L ( z ) | \leq - \ln ( 1 - | z | )$ for all $z$ in $D$. Deduce that the series $\sum _ { n \geq 1 } L \left( z ^ { n } \right)$ is convergent for all $z$ in $D$.

Show that $|L(z)| \leq -\ln(1-|z|)$ for all $z$ in $D$. Deduce that the series $\sum_{n \geq 1} L(z^n)$ is convergent for all $z$ in $D$.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
For $x \in \mathbb{R} \backslash \mathbb{Z}$, justify that the series defining $g(x)$ is convergent.

For $x \in \mathbb{R} \backslash \mathbb{Z}$, justify that the series defining $g(x)$ is convergent, where $$g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right)$$

Show that for all $\theta \in ] - \pi ; \pi [$, the function $f$ defined by
$$\begin{aligned} f : ] 0 ; + \infty [ & \longrightarrow \mathbf { C } \\ t & \longmapsto \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \end{aligned}$$
is defined and integrable on $] 0 ; + \infty [$, where $x$ is a fixed element of $]0;1[$.

Justify that the series $\sum_{k \geqslant 2} \frac{\ln(k)}{k(k-1)}$ converges.

For all real $t \geqslant 2$, we define $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Justify that the function $t \mapsto \frac{R(t)}{t(\ln(t))^{2}}$ is integrable on $[2, +\infty[$.

Justify that the series $\sum_{k \geqslant 2} \frac{\ln(k)}{k(k-1)}$ converges.

We set, for all real $t \geqslant 2$, $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Justify that the function $t \mapsto \frac{R(t)}{t(\ln(t))^2}$ is integrable on $[2, +\infty[$.