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Source. Theorem 1 (Теорема 1), printed p. 66 of the Russian original (physical PDF p. 2), read on the page image. The proof begins in section 2 (p. 69), which reduces Theorems 1 and 2 to the zero estimate Theorem 3 (p. 69); it was not read here.

Statement

For every $q\in\mathbb C$ with $0<|q|<1$, among the numbers $q$, $P(q)$, $Q(q)$, $R(q)$ there are at least three algebraically independent over $\mathbb Q$. Here (p. 65)

$$P(z)=1-24\sum_{n=1}^{\infty}\sigma_1(n)z^n,\quad Q(z)=1+240\sum_{n=1}^{\infty}\sigma_3(n)z^n,\quad R(z)=1-504\sum_{n=1}^{\infty}\sigma_5(n)z^n,$$

with $\sigma_k(n)=\sum_{d\mid n}d^k$, are Ramanujan's functions (the Eisenstein series $E_2,E_4,E_6$ in the variable $z=e^{2\pi i\tau}$); they satisfy $\theta P=(P^2-Q)/12$, $\theta Q=(PQ-R)/3$, $\theta R=(PR-Q^2)/2$ with $\theta=z\,d/dz$ (formula (1)). Equivalently, the field $\mathbb Q(q,P(q),Q(q),R(q))$ has transcendence degree at least $3$, the form in which Waldschmidt's exposé restates the theorem (Théorème 4).

Proof pointer

Ingredients named in sections 1 and 2 and in the expositions: Mahler's 1969 theorem that $P,Q,R$ are algebraically independent over $\mathbb C(z)$; the differential system (1); the zero estimate Theorem 3 (p. 69), which bounds $\operatorname{ord}_{z=0}A(z,P(z),Q(z),R(z))$ by $2\cdot10^{45}L_1L_2^3$ for nonzero $A$ with $\deg_zA\le L_1$, $\deg_{x_i}A\le L_2$; an auxiliary polynomial (Lemma 2.1, p. 69); and Philippon's algebraic independence criterion (his Theorem 2.11, which gives Lemma 2.5, p. 75). Expositions: Waldschmidt, Séminaire Bourbaki exposé 824 (1997), section 2.2 (p. 118) for the statement and section 2.5 (from p. 126) for the proof; the Lecture Notes in Mathematics 1752 chapter (not read). None of the proof was checked here.

Consequence used in the corpus

Corollary 2 (pp. 66--67): for algebraic $q$, $P(q),Q(q),R(q)$ are algebraically independent, in particular transcendental; at $q=1/2$ this gives the transcendence of $\sum_{n\ge1}\sigma(n)/2^n$.

Coverage

Statement read on the page image; proof not read. Relied on as accepted literature: refereed (Mat. Sb.), Zbl 0898.11031, expounded in the Bourbaki exposé of November 1996, and followed by the 1997 Ostrowski Prize to Nesterenko.

Bears on. #250, through Corollary 2.