Nesterenko 1996 modular functions transcendence questions

../

corollary_2: States that for algebraic q with 0 < |q| < 1 the numbers P(q), Q(q), R(q) are algebraically independent, hence transcendental; at q = 1/2 this makes the sum of sigma(n) over 2^n transcendental.

corollary_5: States that each of the triples pi, e^pi, Gamma(1/4) and pi, e^(pi sqrt 3), Gamma(1/3) consists of numbers algebraically independent over the rationals, so in particular pi and e^pi are algebraically independent.

theorem_1: States that for every complex q with 0 < |q| < 1 at least three of the numbers q, P(q), Q(q), R(q), built from Ramanujan's functions, are algebraically independent over the rationals.

theorem_2: States that when q, P(q), Q(q), R(q) are algebraic over the field generated by three complex numbers, every nonzero integer polynomial A, evaluated at those three numbers, has modulus above exp(-gamma_1 t(A)^4 ln^24 t(A)).

theorem_3: States that a nonzero polynomial of degree at most L_1 in z and at most L_2 in each other variable, evaluated at z, P(z), Q(z), R(z), vanishes at z = 0 to order at most 2*10^45 L_1 L_2^3.

---

Yu. V. Nesterenko, Modular functions and transcendence questions (Russian: Модулярные функции и вопросы трансцендентности), Mat. Sb. 187 (1996), no. 9, 65--96; English translation Sb. Math. 187 (1996), no. 9, 1319--1348, DOI 10.1070/SM1996v187n09ABEH000158; Zbl 0898.11031 (reviewer J. Wolfart); MR1422383. Received by the editors 7 March 1996 (p. 96). UDC 511.36.

The copy read for this card is the Russian original from mathnet.ru (record https://www.mathnet.ru/eng/sm158): 32 physical pages, printed pp. 65--96 (physical p. $n$ is printed p. $64+n$). Its text layer is unusable (font encoding), so the statements below were read on the page images. Provenance: fetched from https://www.mathnet.ru/php/getFT.phtml?jrnid=sm&paperid=158&what=fullt&option_lang=eng on 2026-09-17 (UTC), 422,932 bytes. The English translation was not read (no free copy); it shares the result labels but not the page numbers, and its pages for the individual theorems are not known here. Result citations use the Russian pagination. The file prints "© Ю. В. Нестеренко 1996" in the footer of its first page (read on the page image; the text layer is garbled) and no license wording on its 32 pages; the Math-Net.Ru Terms of Use (https://www.mathnet.ru/php/agreement.phtml?option_lang=eng, read 2026-10-02) state that "All materials published on this website including full-text articles, abstracts and author indexes are fully copyrighted by Steklov Mathematical Institute, Russian Academy of Sciences, and/or by other copyright holder" and that "Reproduction or republication of the materials contained on Math-Net.Ru in any form requires written permission of the copyright holder", allowing printing for noncommercial teaching or research only and naming no open license, every other right reserved.

Announcement version. The site's reference [Ne96] for Problem 250 is the C. R. note: Yu. V. Nesterenko, Modular functions and transcendence problems, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 10, 909--914, Zbl 0859.11047 (reviewer F. Gramain). It was not read here (no free copy; only its zbMATH record was read), its own labels are unknown, and the Mat. Sb. paper's nineteen-item bibliography does not cite it. Waldschmidt's Bourbaki exposé (card) cites the two as [2] and [3] and presents the theorem as "le résultat principal de [2] et [3]" (p. 118).

Contents of section 1 (pp. 65--69)

Ramanujan's functions (p. 65)

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

$\sigma_k(n)=\sum_{d\mid n}d^k$, satisfy (1) $\theta P=(P^2-Q)/12$, $\theta Q=(PQ-R)/3$, $\theta R=(PR-Q^2)/2$, $\theta=z\,d/dz$. Mahler (1969) proved $P,Q,R$ algebraically independent over $\mathbb C(z)$ (p. 66); $E_4(\tau)=Q(e^{2\pi i\tau})$ and $E_6(\tau)=R(e^{2\pi i\tau})$ are modular forms of weights 4 and 6, and $E_2(\tau)=P(e^{2\pi i\tau})$ has some modular properties.

• Theorem 1 (p. 66): for every $q\in\mathbb C$ with $0<|q|<1$, at least three of $q,P(q),Q(q),R(q)$ are algebraically independent over $\mathbb Q$.
• Corollary 1 (p. 66): with $\Delta=(Q^3-R^2)/1728$ and $J=Q^3/\Delta=1/z+744+\sum c(n)z^n$, for every $\tau$ with $\operatorname{Im}\tau>0$ not congruent under the modular group to $i$ or $\zeta=e^{2\pi i/3}$, and $q=e^{2\pi i\tau}$, each of $\{q,J(q),J'(q),J''(q)\}$ and $\{q,j(\tau),\pi^{-1}j'(\tau),\pi^{-2}j''(\tau)\}$ contains at least three algebraically independent numbers.
• Corollary 2 (pp. 66--67): for algebraic $q$ with $0<|q|<1$, each of $\{P(q),Q(q),R(q)\}$ and $\{J(q),\theta J(q),\theta^2J(q)\}$ consists of algebraically independent numbers; in particular each is transcendental.
• Corollaries 3--6 (pp. 67--68): for a Weierstrass $\wp$ with algebraic invariants, periods $\omega_1,\omega_2$ with $\operatorname{Im}(\omega_2/\omega_1)\ne0$ and the quasi-period $\eta_1$ of $\omega_1$, the numbers $e^{2\pi i\omega_2/\omega_1},\omega_1/\pi,\eta_1/\pi$ are algebraically independent; with complex multiplication by a field $k$, so are $\{\pi,\omega,e^{2\pi i\tau}\}$ and $\{\omega,\eta,e^{2\pi i\tau}\}$ for every period $\omega$, its quasi-period $\eta$ and every $\tau\in k$ with $\operatorname{Im}\tau\ne0$; hence $\{\pi,e^{\pi},\Gamma(1/4)\}$ and $\{\pi,e^{\pi\sqrt3},\Gamma(1/3)\}$ are algebraically independent (Corollary 5, p. 68), and $\pi$ and $e^{\pi\sqrt D}$ for every natural $D$ (Corollary 6).
• Theorem 2 (p. 69): a measure of algebraic independence. For $q\in\mathbb C$, $0<|q|<1$, and $\theta_1,\theta_2,\theta_3\in\mathbb C$ such that $q,P(q),Q(q),R(q)$ are algebraic over $\mathbb Q(\theta_1,\theta_2,\theta_3)$, there is $\gamma_1$ depending only on $q$ and the $\theta_i$ with $|A(\theta_1,\theta_2,\theta_3)|>\exp(-\gamma_1t(A)^4\ln^{24}t(A))$ for every nonzero $A\in\mathbb Z[x_1,x_2,x_3]$, $t(A)=\ln H(A)+\deg A$. (The sharper measure with $\ln^9$ that Zudilin 2002 quotes as "Nesterenko's Theorem 2" (Sb. Math. p. 1152) is cited there to Nesterenko's 1997 Steklov Institute paper, not to this one.)
• Theorem 3 (p. 69): the zero estimate. For integers $L_1,L_2\ge1$ and nonzero $A\in\mathbb C[z,x_1,x_2,x_3]$ with $\deg_zA\le L_1$, $\deg_{x_i}A\le L_2$, $\operatorname{ord}_{z=0}A(z,P(z),Q(z),R(z))\le cL_1L_2^3$ with $c=2\cdot10^{45}$. Section 2 (from p. 69) reduces Theorems 1 and 2 to Theorem 3, starting from Lemma 2.1: for every sufficiently large $N$ there is a nonzero $A\in\mathbb Z[z,x_1,x_2,x_3]$ of degree at most $N$ in $z$ and in each $x_i$, with $\ln H(A)\le85N\ln N$, such that $A(z,P(z),Q(z),R(z))$ vanishes at $z=0$ to order at least $[(N+1)^4/2]$.
• Historical remarks (p. 68): Mahler's 1969 conjecture on $J$ was proved in 1995 by Barré-Sirieix, Diaz, Gramain and Philibert; Chudnovsky proved the algebraic independence of $\pi,\Gamma(1/4)$ in 1976; Bertrand's 1977 conjecture is proved by Corollary 2; the independence of $\pi$ and $e^{\pi}$ (Corollary 5) was an old folklore conjecture.

Compiled scope

Theorem 1 and Corollary 2 were read on the page images (pp. 66--67) and are recorded with the specialization to $q=1/2$. Theorems 2 and 3 (p. 69) and Corollary 5 (p. 68) were read on the page images and have result pages of their own; Corollaries 1, 3, 4 and 6 are summarized above only. The proofs (sections 2 onward) were not read; the theorem is relied on as accepted literature (refereed, reviewed in zbMATH, expounded in the Bourbaki exposé and in Lecture Notes in Mathematics 1752, and followed by the 1997 Ostrowski Prize to Nesterenko). No reconstruction is planned here.

Bears on. #250: Corollary 2 at $q=1/2$ gives the transcendence, hence the irrationality, of $\sum_{n\ge1}\sigma(n)/2^n=(1-P(1/2))/24$; the site's [Ne96] is the C. R. announcement of this result. Theorem 1 is the general statement Corollary 2 specializes, and Theorem 3 is the zero estimate the proof of Theorem 1 rests on; Theorem 2 and Corollary 5 bear on no problem in the corpus.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.