Je travaille sur la constante d'Euler-Mascheroni γ et je cherche à confirmer une information. Est-ce que quelqu'un a prouvé que cette constante est algébrique ou transcendante ?
Properties

The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10. The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics.

However, some progress has been made. Kurt Mahler showed in 1968 that the number π 2 Y 0 (2) J 0 (2) − γ {\displaystyle {\frac {\pi }{2}}{\frac {Y_{0}(2)}{J_{0}(2)}}-\gamma } is transcendental (here, J α (x) {\displaystyle J_{\alpha }(x)} and Y α (x) {\displaystyle Y_{\alpha }(x)} are Bessel functions). In 2009 Alexander Aptekarev proved that at least one of Euler's constant γ and the Euler–Gompertz constant δ is irrational; Tanguy Rivoal proved in 2012 that at least one of them is transcendental. In 2010 M. Ram Murty and N. Saradha showed that at most one of the numbers of the form

γ (a, q) = lim n → ∞ ((∑ k = 0 n 1 a + k q) − log (a + n q) q) {\displaystyle \gamma (a,q)=\lim _{n\rightarrow \infty }\left(\left(\sum _{k=0}^{n}{\frac {1}{a+kq}}\right)-{\frac {\log {(a+nq})}{q}}\right)}

with q ≥ 2 and 1 ≤ a < q is algebraic; this family includes the special case γ (2,4) = γ / 4. In 2013 M. Ram Murty and A. Zaytseva found a different family containing γ, which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.
La constante d'Euler-Mascheroni γ

La constante d'Euler-Mascheroni est définie comme la limite suivante :

La constante d'Euler-Mascheroni apparaît dans le troisième théorème de Mertens et a des relations avec la fonction gamma, la fonction zêta et de nombreuses intégrales et séries différentes.

Il est encore inconnu si γ est rationnel ou non.

La valeur numérique de γ est approximativement de 0,57721.
### Query ###
Je travaille sur la constante d'Euler-Mascheroni γ et je cherche à confirmer une information. Est-ce que quelqu'un a prouvé que cette constante est algébrique ou transcendante ?

### Main Text ###
Properties

The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10. The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics.

However, some progress has been made. Kurt Mahler showed in 1968 that the number π 2 Y 0 (2) J 0 (2) − γ {\displaystyle {\frac {\pi }{2}}{\frac {Y_{0}(2)}{J_{0}(2)}}-\gamma } is transcendental (here, J α (x) {\displaystyle J_{\alpha }(x)} and Y α (x) {\displaystyle Y_{\alpha }(x)} are Bessel functions). In 2009 Alexander Aptekarev proved that at least one of Euler's constant γ and the Euler–Gompertz constant δ is irrational; Tanguy Rivoal proved in 2012 that at least one of them is transcendental. In 2010 M. Ram Murty and N. Saradha showed that at most one of the numbers of the form

γ (a, q) = lim n → ∞ ((∑ k = 0 n 1 a + k q) − log (a + n q) q) {\displaystyle \gamma (a,q)=\lim _{n\rightarrow \infty }\left(\left(\sum _{k=0}^{n}{\frac {1}{a+kq}}\right)-{\frac {\log {(a+nq})}{q}}\right)}

with q ≥ 2 and 1 ≤ a < q is algebraic; this family includes the special case γ (2,4) = γ / 4. In 2013 M. Ram Murty and A. Zaytseva found a different family containing γ, which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.

### Supplementary Text ###
La constante d'Euler-Mascheroni γ

La constante d'Euler-Mascheroni est définie comme la limite suivante :

La constante d'Euler-Mascheroni apparaît dans le troisième théorème de Mertens et a des relations avec la fonction gamma, la fonction zêta et de nombreuses intégrales et séries différentes.

Il est encore inconnu si γ est rationnel ou non.

La valeur numérique de γ est approximativement de 0,57721.

### Draft ###

### 1. Query Parsing

"Je travaille sur la constante d'Euler-Mascheroni γ" → User working on γ constant
"quelqu'un a prouvé que cette constante est algébrique ou transcendante" → Seeks proof status

Key question: Is γ algebraic or transcendental?

### 2. Mathematical Context Retrieval

Euler-Mascheroni constant γ = lim(n→∞)[H_n - ln(n)] where H_n = Σ(1/k)

Historical status check:
● Neither algebraic nor transcendental proven
◐ Irrationality unknown - major open problem

### 3. Known Results Compilation

**Transcendence connections:**
- π²/2[Y₀(2)/J₀(2)] - γ transcendental (Mahler 1968)
- At least one of {γ, δ} transcendental (Rivoal 2012)

**Algebraicity bounds:**
- If γ rational → denominator > 10¹⁰ (Papanikolaou 1997)
- Family γ(a,q) = lim[Σ(1/(a+kq)) - ln(a+nq)/q] → at most one algebraic (Murty-Saradha 2010)

### 4. Current State Assessment

⟨H≈0.8⟩ The constant sits in this strange limbo... Euler knew it well, worked with it extensively, yet its fundamental nature remains mysterious. Like π and e, it emerged from analysis, not from the classical constructions that give rise to algebraic numbers. But unlike those two, no transcendence proof exists.

The continued fraction work shows it's not "obviously" rational. The Bessel function connection suggests transcendental behavior. Yet both possibilities remain live.

### 5. Response Structure

Answer: Neither proven
- Historical context: Euler-Mascheroni era → modern uncertainty
- Current status: Open problem
- Key results: Continued fraction bounds, Bessel transcendence, family analysis
- Conclusion: Active research area, no definitive answer yet

∴ Honest response acknowledging ongoing mathematical mystery

**Verification needed:**
☐ Check latest research post-2013
☐ Confirm no breakthrough announcements
☑ Provide appropriate uncertainty language
☑ Balance technical accuracy with accessibility

**Key insight:** This exemplifies how major constants can remain fundamentally unknown despite centuries of study.

### Answer###
La constante d'Euler-Mascheroni γ n'a pas encore été prouvée algébrique ni transcendante. En réalité, on ignore même si γ est irrationnel. Des progrès ont néanmoins été réalisés dans cette direction.

L'analyse des fractions continues a montré que si γ est rationnel, son dénominateur doit être supérieur à 10¹⁰. Concernant la transcendance, Kurt Mahler a démontré en 1968 que le nombre π²/2 × Y₀(2)/J₀(2) - γ est transcendantal, où Y₀ et J₀ sont des fonctions de Bessel. Plus récemment, en 2012, Tanguy Rivoal a prouvé qu'au moins l'une des constantes γ et δ d'Euler-Gompertz est transcendante.

Certains résultats se rapprochent de la question en étudiant des familles de nombres apparentés. Par exemple, au moins l'un des nombres de la forme γ(a,q) = lim((Σ(1/(a+kq))) - ln(a+nq)/q) avec q ≥ 2 et 1 ≤ a < q est algébrique.

Cette situation fait de l'irrationalité de γ une question ouverte majeure en mathématiques, car γ apparaît dans un grand nombre d'équations et d'intégrales.