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Statement

Theorem 2 (p. 69). Let $q\in\mathbb C$ with $0<|q|<1$, and let $\theta_1,\theta_2,\theta_3\in\mathbb C$ be such that all of the numbers $q,P(q),Q(q),R(q)$ are algebraic over the field $\mathbb Q(\theta_1,\theta_2,\theta_3)$. Then there is a constant $\gamma_1$, depending only on $q$ and the $\theta_i$, such that for every polynomial $A\in\mathbb Z[x_1,x_2,x_3]$, $A\ne0$,

$$|A(\theta_1,\theta_2,\theta_3)|>\exp\bigl(-\gamma_1\,t(A)^4\ln^{24}t(A)\bigr),$$

where $t(A)=\ln H(A)+\deg A$ and $H(A)$ is the largest modulus of the coefficients of $A$. The paper adds that in particular such a bound holds for each of the triples $\pi,e^{\pi},\Gamma(1/4)$ and $\pi,e^{\pi\sqrt3},\Gamma(1/3)$.

Here $P,Q,R$ are Ramanujan's functions of Theorem 1. The hypothesis forces $\theta_1,\theta_2,\theta_3$ to be algebraically independent, by Theorem 1; the theorem is the quantitative form of that statement. The paper introduces it on p. 68 with the remark that the method of proof of Theorem 1 readily yields quantitative results.

Source. Yu. V. Nesterenko, Modular functions and transcendence questions, Mat. Sb. 187 (1996), no. 9, 65--96 (Russian; English translation Sb. Math. 187 (1996), no. 9, 1319--1348), Theorem 2 (Теорема 2), p. 69 of the Russian original; see the source card. Labels and pages follow the Russian pagination.

Read depth. Claims checked: the statement was read clause by clause on the page image. The proof was not read.

Proof pointer

Section 2 (from p. 69) reduces Theorems 1 and 2 to the zero estimate Theorem 3. On p. 75 the paper says that Theorem 2 comes from replacing the algebraic independence criterion used for Theorem 1 (Lemma 2.5) by a criterion of M. Ably (J. Number Theory 42 (1992), 194--231), which yields an intermediate bound for ideals (Theorem 4, p. 75). None of this was checked here.

Bears on

No Erdős problem in the corpus.