Centrale-Supélec 2026 — Mathématiques 2 (MP, MPI)

Independent solutions, written by Pulsars. Neither official nor affiliated with Concours Centrale-Supélec.

1. Exercise 1 — Partie A · I — La fonction ζ sur ]1, +∞[ (Q1 à Q3)

   Let u_k(s) = 1/k^s and ζ(s) = Σ_{k⩾1} 1/k^s.

   • Q1 — does the series of functions converge normally on ]1, +∞[?
   • Q2 — continuity of ζ on this interval.
   • Q3 — ζ does not vanish there; determine its sign.
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2. Exercise 2 — Partie A · II — Cas de la variable complexe (Q4 à Q10)

   s is now complex, with real part x = Re(s) > 1, and (p_i) denotes the increasing sequence of prime numbers.

   • Q4 — modulus of t^s for t > 0.
   • Q5 — absolute convergence of the series defining ζ(s).
   • Q6 — summability of the family indexed by the exponents, and the partial Euler product S_n(s).
   • Q7 — limit of S_n(s) as n → +∞.
   • Q8 — nature of the series Σ 1/p_i^x.
   • Q9 — convergence of the sequence a_n = Π p_i^x/(p_i^x + 1) to a real number > 0.
   • Q10 — the lower bound |S_n(s)| ⩾ a_n, hence ζ(s) ≠ 0.
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3. Exercise 3 — Partie A · III — Un calcul d'intégrale (Q11)

   • Q11 — integrability of t ↦ t^{2(s−1)} on ]0, 1] for Re(s) > 1/2, then computation of ∫_0^1 |t^{s−1}|² dt.
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4. Exercise 4 — Partie A · IV — Une étude géométrique (Q12 à Q14)

   For d ∈ ]0,1[, Δ_d denotes the set of points with affix s satisfying d²|s|² ⩾ 2 Re(s) − 1.

   • Q12 — Δ_d is the complement of a disc, whose centre and radius are to be determined.
   • Q13 — image of ]0,1[ under g(d) = 1/d² − √(1−d²)/d².
   • Q14 — the intersection of all the Δ_d.
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5. Exercise 5 — Partie B · I — Transformée de Mellin des fonctions f_k (Q15 à Q21)

   For k ⩾ 2, f_k(t) = (1/k)⌊1/t⌋ − ⌊1/(kt)⌋ on ]0,1].

   • Q15 — values of f_k(1/i) for 1 ⩽ i ⩽ k.
   • Q16 — the bounds 0 ⩽ f_k(t) ⩽ (k−1)/k.
   • Q17 — the set of values taken by f_k.
   • Q18 — piecewise continuity and integrability.
   • Q19 to Q21 — convergence of the associated integral, computation of F(s) = ∫_0^1 ⌊1/t⌋ t^{s−1} dt, then the transform of f_k.
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6. Exercise 6 — Partie B · II — Prolongement de la fonction ζ (Q22 à Q24)

   Here s denotes a strictly positive real number, and h(t) = 1/t^s.

   • Q22 — maximum of |h′| on [k, k+1].
   • Q23 — convergence of the series Σ ∫_k^{k+1}(1/k^s − 1/t^s)dt, and the bounds 0 ⩽ G(s) ⩽ s·ζ(s+1).
   • Q24 — the identity ζ(s) − G(s) = 1/(s−1) for s > 1, then the behaviour of ζ near 1.
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7. Exercise 7 — Partie B · III — Intégrale des fonctions f_k (Q25, Q26)

   • Q25 — passing to the limit s → 1^+ in ∫_0^1 f_k(t)t^{s−1}dt.
   • Q26 — the identity ∫_0^1 f_k(t)dt = ln(k)/k.
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8. Exercise 8 — Partie C · I — Une forme bilinéaire symétrique positive (Q27 à Q29)

   E denotes the space of piecewise continuous functions on ]0,1], and L² the subset of those whose square is integrable.

   • Q27 — three statements to decide: bounded functions, the equality E = L², and integrable functions.
   • Q28 — absolute convergence of ∫ fg and vector subspace structure.
   • Q29 — (f|g) = ∫_0^1 fg is a positive symmetric bilinear form, but not an inner product on L².
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9. Exercise 9 — Partie C · II — Un espace préhilbertien réel (Q30 à Q33)

   Assume that Σ_{k=2}^{n} λ_k f_k = 0 on ]0,1].

   • Q30 — by evaluating on ]1/2, 1], obtain Σ λ_k/k = 0.
   • Q31 — by evaluating at a well-chosen point, a relation for each ℓ.
   • Q32 — deduce that λ_2 = 0, then that all the λ_k are zero: what property of the family (f_k) has just been established?
   • Q33 — (.|.) is an inner product on F = Vect(1, f_2, f_3, …).
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10. Exercise 10 — Partie C · III — Distance d'un vecteur à des sous-espaces (Q34 à Q36)

    We work in F equipped with the previous inner product, and d_n denotes the distance from the constant function 1 to the subspace spanned by f_2, …, f_n.

    • Q34 — existence and uniqueness of an orthonormal family (e_k) adapted to the nested subspaces, with a sign condition.
    • Q35 — the identity d_n² = 1 − Σ (1|e_k)².
    • Q36 — convergence of the sequence (d_n).
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11. Exercise 11 — Partie D — Synthèse (Q37, Q38)

    The Cauchy-Schwarz inequality is admitted for piecewise continuous, square-integrable, complex-valued functions.

    • Q37 — justify its application to 1 − Σ λ_k f_k and to t^{s−1}.
    • Q38 — establish d_n²|s|² ⩾ 2 Re(s) − 1, then conclude when d_n → 0.
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