grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2020 centrale-maths1__mp

42 maths questions

Q1 Groups Binary Operation Properties View
Verify that $\delta$ is a neutral element for the operation $*$.
Q2 Number Theory Arithmetic Functions and Multiplicative Number Theory View
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 $\mathcal{C}_n = \left\{ \left( d_1, d_2 \right) \in \left( \mathbb{N}^* \right)^2 \mid d_1 d_2 = n \right\}$.
Q3 Groups Binary Operation Properties View
Deduce that $*$ is commutative.
Q4 Number Theory Binary Operation Properties View
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.
Q5 Groups Ring and Field Structure View
What can be said about $(\mathbb{A}, +, *)$?
Q6 Number Theory Arithmetic Functions and Multiplicative Number Theory View
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 $f = g$.
Q7 Number Theory GCD, LCM, and Coprimality View
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 $\mathcal{D}_{mn}$.
Q8 Number Theory Arithmetic Functions and Multiplicative Number Theory View
Deduce that if $f$ and $g$ are two multiplicative functions, then $f * g$ is still multiplicative.
Q9 Number Theory Arithmetic Functions and Multiplicative Number Theory View
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 $f * g = \delta$.
Q10 Groups Algebraic Structure Identification View
What can be said about the set $\mathcal{M}$ equipped with the operation $*$?
Q11 Number Theory Arithmetic Functions and Multiplicative Number Theory View
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.
Q12 Number Theory Arithmetic Functions and Multiplicative Number Theory View
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 $\mu * \mathbf{1} = \delta$.
Q13 Number Theory Arithmetic Functions and Multiplicative Number Theory View
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 $n \in \mathbb{N}^*$,
$$f(n) = \sum_{d \mid n} \mu(d) F\left(\frac{n}{d}\right)$$
Q14 Number Theory Arithmetic Functions and Multiplicative Number Theory View
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 $\varphi = \mu * \mathbf{I}$.
Q15 Number Theory Matrix Algebra and Product Properties View
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 $D$.
Q16 Number Theory Determinant and Rank Computation View
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 $m_{ij}' = \begin{cases} g(j) & \text{if } j \text{ divides } i, \\ 0 & \text{otherwise.} \end{cases}$
Using the result $M = M' D^\top$, deduce that the determinant of $M$ equals
$$\operatorname{det} M = \prod_{k=1}^{n} g(k)$$
Q17 Number Theory Convergence/Divergence Determination of Numerical Series View
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.
Q18 Number Theory Evaluation of a Finite or Infinite Sum View
Let $f$ and $g$ be two arithmetic functions with finite abscissas of convergence. Show that if, for all $s > \max\left(A_c(f), A_c(g)\right), L_f(s) = L_g(s)$, then $f = g$.
Q19 Number Theory Properties and Manipulation of Power Series or Formal Series View
Let $f$ and $g$ be two arithmetic functions with finite abscissas of convergence. Show that, for all $s > \max\left(A_c(f), A_c(g)\right)$,
$$L_{f*g}(s) = L_f(s) L_g(s)$$
Q20 Groups Diagonalizability and Similarity View
For all permutations $\rho, \rho' \in \mathfrak{S}_n$, show that $P_{\rho\rho'} = P_\rho P_{\rho'}$. Deduce that, for all permutations $\sigma, \tau \in \mathfrak{S}_n$, if $\sigma$ and $\tau$ are conjugate then $P_\sigma$ and $P_\tau$ are similar.
Q21 Groups Symmetric Group and Permutation Properties View
We consider, in this question only, $n = 7$ and the cycles $\gamma_1 = (1\;3)$ and $\gamma_2 = (2\;6\;4)$. We also consider a permutation $\rho \in \mathfrak{S}_7$ such that $\rho(1) = 2, \rho(3) = 6$ and $\rho(7) = 4$. Verify that $\rho \gamma_1 \rho^{-1} = \gamma_2$.
Q22 Groups Symmetric Group and Permutation Properties View
Show that, in $\mathfrak{S}_n$, two cycles of the same length are conjugate.
Q23 Groups Symmetric Group and Permutation Properties View
Show that $\sigma \in \mathfrak{S}_n$ and $\tau \in \mathfrak{S}_n$ are conjugate if and only if, for all $\ell \in \llbracket 1, n \rrbracket, c_\ell(\sigma) = c_\ell(\tau)$, where for $\ell \in \llbracket 2, n \rrbracket$, $c_\ell(\sigma)$ denotes the number of cycles of length $\ell$ in the decomposition of $\sigma$ into cycles with disjoint supports, and $c_1(\sigma) = \operatorname{Card}\{j \in \llbracket 1, n \rrbracket, \sigma(j) = j\}$.
Q24 Groups Determinant and Rank Computation View
Let $\ell \in \llbracket 2, n \rrbracket$ and let $\gamma \in \mathfrak{S}_\ell$ be a cycle of length $\ell$. Show that $\chi_\gamma(X) = X^\ell - 1$.
One may reduce to the case $\gamma = (12\cdots\ell)$ and consider the matrix
$$\Gamma_\ell = \left( \begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & 1 \\ 1 & 0 & \cdots & \cdots & 0 & 0 \\ 0 & 1 & \ddots & & \vdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & 0 \\ 0 & \cdots & \cdots & 0 & 1 & 0 \end{array} \right) \in \mathcal{M}_\ell(\mathbb{C})$$
Q25 Groups Diagonalizability and Similarity View
Show that if $\sigma \in \mathfrak{S}_n$, then $\chi_\sigma(X) = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell(\sigma)}$.
One may justify that $P_\sigma$ is similar to a block diagonal matrix whose blocks are matrices of the form $\Gamma_\ell$ ($\ell \geq 1$), where $\Gamma_\ell$ is defined above if $\ell \geq 2$ and where $\Gamma_\ell = (1)$ if $\ell = 1$.
Q26 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
By reasoning on the multiplicity of the roots of $\chi_\sigma$ and $\chi_\tau$, show that if $P_\sigma$ and $P_\tau$ are similar, then, for all $q \in \llbracket 1, n \rrbracket$,
$$\sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\sigma) = \sum_{\substack{\ell=1 \\ q \mid \ell}}^{n} c_\ell(\tau)$$
(We sum over the values of $\ell$ that are multiples of $q$ and belong to $\llbracket 1, n \rrbracket$.)
Q27 Number Theory Diagonalizability and Similarity View
Deduce property (S): The permutation matrices $P_\sigma$ and $P_\tau$ are similar if and only if the permutations $\sigma$ and $\tau$ are conjugate.
One may compute $T_\sigma D$ where $T_\sigma$ is the cycle type of $\sigma$ and $D$ is the divisor matrix defined in I.D.
Q28 Matrices Linear Transformation and Endomorphism Properties View
Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Show that $u$ is a permutation endomorphism if and only if there exists a basis in which its matrix is a permutation matrix.
Q29 Invariant lines and eigenvalues and vectors Diagonalizability and Similarity View
Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be a permutation endomorphism of $E$. Show that $u$ is diagonalizable and that its trace belongs to $\llbracket 0, n \rrbracket$.
Q30 Invariant lines and eigenvalues and vectors Diagonalizability and Similarity View
Let $A, B$ be two diagonalizable matrices of $\mathcal{M}_n(\mathbb{C})$. Show that $A$ and $B$ are similar if and only if they have the same characteristic polynomial.
Q31 Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be an endomorphism of $E$ such that $u^2 = \operatorname{Id}_E$. Show that $u$ is a permutation endomorphism if and only if $\operatorname{Tr}(u)$ is a natural integer.
Q32 Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Study whether the equivalence of the previous question holds when we replace the hypothesis $u^2 = \operatorname{Id}_E$ by $u^k = \operatorname{Id}_E$ for $k = 3$, then for $k = 4$.
Q33 Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be an endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if it satisfies the following two conditions:
(a) there exist natural integers $c_1, \ldots, c_n$ such that $\chi_u = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell}$;
(b) there exists $N$ such that $u^N = \operatorname{Id}_E$.
Q34 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ and $v$ be two endomorphisms of $E$ such that, for all $k \in \mathbb{N}, \operatorname{Tr}\left(u^k\right) = \operatorname{Tr}\left(v^k\right)$. Show that $u$ and $v$ have the same characteristic polynomial.
Q35 Invariant lines and eigenvalues and vectors Matrix Power Computation and Application View
Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be a diagonalizable endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if there exist natural integers $c_1, \ldots, c_n$ such that, for all $k \in \mathbb{N}$,
$$\operatorname{Tr}\left(u^k\right) = \sum_{\substack{\ell=1 \\ \ell \mid k}}^{n} \ell c_\ell$$
(We sum over the values of $\ell$ dividing $k$ and belonging to $\llbracket 1, n \rrbracket$.)
Q36 Invariant lines and eigenvalues and vectors Arithmetic Functions and Multiplicative Number Theory View
We define the Redheffer matrix by $H_n = \left(h_{ij}\right)_{(i,j) \in \llbracket 1,n \rrbracket^2}$ where
$$h_{ij} = \begin{cases} 1 & \text{if } j = 1 \\ 1 & \text{if } i \text{ divides } j \text{ and } j \neq 1 \\ 0 & \text{otherwise.} \end{cases}$$
We also define the Mertens function $M$, by setting, for all $n \in \mathbb{N}^*, M(n) = \sum_{k=1}^{n} \mu(k)$ where $\mu$ is the Möbius function.
Let $A_n = \left(a_{ij}\right)_{(i,j) \in \llbracket 1,n \rrbracket^2}$ be the matrix with general term
$$a_{ij} = \begin{cases} \mu(j) & \text{if } i = 1 \\ 1 & \text{if } i = j \\ 0 & \text{otherwise} \end{cases}$$
and $C_n = A_n H_n$. By computing the coefficients of $C_n$, show that $\operatorname{det} H_n = M(n)$.
For the computation of the term with index $(i,j)$ of $C_n$, one may distinguish the case $i = j = 1$, the case $i = 1, j > 1$ and the case $i > 1, j > 1$.
Q37 Number Theory Arithmetic Functions and Multiplicative Number Theory View
We define the Redheffer matrix by $H_n = \left(h_{ij}\right)_{(i,j) \in \llbracket 1,n \rrbracket^2}$ where
$$h_{ij} = \begin{cases} 1 & \text{if } j = 1 \\ 1 & \text{if } i \text{ divides } j \text{ and } j \neq 1 \\ 0 & \text{otherwise.} \end{cases}$$
We denote $\chi_n$ the characteristic polynomial of $H_n$. For $\lambda$ real distinct from 1, we define by recursion the arithmetic function $\mathbf{b}$, by setting $\mathbf{b}(1) = 1$ and, for all natural integer $j \geq 2$,
$$\mathbf{b}(j) = \frac{1}{\lambda - 1} \sum_{d \mid j, d \neq j} \mathbf{b}(d)$$
We also define the matrix $B_n(\lambda) = \left(b_{ij}\right)_{(i,j) \in \llbracket 1,n \rrbracket^2}$ with general term
$$b_{ij} = \begin{cases} \mathbf{b}(j) & \text{if } i = 1 \\ 1 & \text{if } i = j \\ 0 & \text{otherwise.} \end{cases}$$
By computing the product $B_n(\lambda)\left(\lambda I_n - H_n\right)$, show that
$$\chi_n(\lambda) = (\lambda - 1)^n - (\lambda - 1)^{n-1} \sum_{j=2}^{n} \mathbf{b}(j).$$
Q38 Number Theory Arithmetic Functions and Multiplicative Number Theory View
In the rest of the problem, we assume that $\lambda$ is a real distinct from 1 and we set $w = \frac{1}{\lambda - 1}$. We further set $\mathbf{f} = (1 + w)\delta - w\mathbf{1}$.
Show that $\mathbf{f} * \mathbf{b} = \delta$.
Q39 Number Theory Evaluation of a Finite or Infinite Sum View
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 $L_{\mathbf{1}}(s)$.
Q40 Number Theory Properties and Manipulation of Power Series or Formal Series View
We assume that $\lambda$ is a real distinct from 1 and we set $w = \frac{1}{\lambda - 1}$. We further set $\mathbf{f} = (1 + w)\delta - w\mathbf{1}$. We denote $\log_2$ the logarithm function in base 2, defined by $\log_2(x) = \frac{\ln(x)}{\ln(2)}$ for all real $x > 0$.
Show that, for $s$ real sufficiently large,
$$\frac{1}{L_{\mathbf{f}}(s)} = 1 + \sum_{m=2}^{\infty} m^{-s} \sum_{k=1}^{\lfloor \log_2 m \rfloor} w^k D_k(m)$$
where $D_k(m)$ is the number of ways to decompose the integer $m$ into a product of $k$ factors greater than or equal to 2, the order of these factors being important.
Q41 Number Theory Evaluation of a Finite or Infinite Sum View
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
$$\chi_n(\lambda) = (\lambda - 1)^n - \sum_{k=1}^{\lfloor \log_2 n \rfloor} (\lambda - 1)^{n-k-1} S_k(n).$$
Q42 Number Theory Compute eigenvalues of a given matrix View
We define the Redheffer matrix $H_n$ and its characteristic polynomial $\chi_n$. We denote $\log_2$ the logarithm function in base 2.
Finally, show that $H_n$ has 1 as an eigenvalue and that its multiplicity is exactly
$$n - \lfloor \log_2 n \rfloor - 1.$$