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On the measure of algebraic independence of the values of Ramanujan functions. (English. Russian original) Zbl 0913.11030

Lupanov, O. B. (ed.), Analytic number theory and applications. Collected papers in honor of the sixtieth birthday of Professor Anatolii Alexeevich Karatsuba. Moscow: MAIK Nauka/Interperiodica Publishing, Proc. Steklov Inst. Math. 218, 294-331 (1997); translation from Tr. Mat. Inst. Steklova 218, 299-334 (1997).
Denote by \(P\), \(Q\), \(R\) (using Ramanujan’s notation) the Eisenstein series \[ P=1-24\sum_{n=1}^{\infty}\sigma_{1}(n)z^{n}, \;Q=1+240\sum_{n=1}^{\infty}\sigma_{3}(n)z^{n}, \quad R=1-504\sum_{n=1}^{\infty}\sigma_{5}(n)z^{n} \] where \(\sigma_{k}(n)=\sum_{d\mid n}d^{k}\). In a previous paper [Yu. V. Nesterenko, Sb. Math. 187, No. 9, 1319-1348 (1996); translation from Mat. Sb. 187, No. 9, 65-96 (1996; Zbl 0898.11031)] the author proved that for any \(q\in\mathbb C\) with \(0<| q| <1\), at least three of the four numbers \(q\), \(P(q)\), \(Q(q)\) and \(R(q)\) are algebraically independent. Here he produces a quantitative refinement as follows. Let \(\omega_{1},\omega_{2},\omega_{3}\) be complex numbers. Assume that each of the four numbers \(q\), \(P(q)\), \(Q(q)\) and \(R(q)\) is algebraic over the field \(\mathbb Q(\omega_{1},\omega_{2},\omega_{3})\). Then there exists a constant \(\gamma\) such that, for any non zero polynomial \(A\in\mathbb Z[X_{1},X_{2},X_{3}]\), if \(H(A)\) denotes the maximum absolute value of the coefficients of \(A\), if \(t(A)=\deg A+\log H(A)\), and if \(S\geq e\) satisfies \(S\geq\log H(A)+(\deg A)\log t(A)\), then \[ | A(\omega_{1},\omega_{2},\omega_{3})| \geq \exp\bigl\{-\gamma S^{4}(\log S)^{9}\bigr\}. \] Similar slightly weaker estimates had been already proved by the author (with the exponent \(9\) of \(\log S\) replaced by \(24\) – op. cit.) and by P. Philippon (with \(9\) replaced by \(12\)) [J. Reine Angew. Math. 497, 1-15 (1998; Zbl 0887.11032)]. As a special case, the author points out that this estimate holds for triples like \((\pi, e^{\pi}, \Gamma(1/4))\) and \((\pi, e^{\pi\sqrt{3}}, \Gamma(1/3))\).
A refinement is given in the special case where \(q\) is algebraic, namely \[ | A(\omega_{1},\omega_{2},\omega_{3})| \geq \exp\bigl\{-\gamma' S(\deg A)^{3} (\log S)^{9}\bigr\}. \] This measure of algebraic independence holds for instance for the following triples, where \(q\) denotes any complex algebraic number with \(0<| q| <1\): \[ \bigl(P(q),\;Q(q),\;R(q)\bigr),\quad \bigl(J(q),\;(qd/dq)J(q),\;(qd/dq)^{2}J(q)\bigr) \] and \[ \sum_{n\geq 1}q^{n^{2}}, \;\sum_{n\geq 1}n^{2}q^{n^{2}}, \;\sum_{n\geq 1}n^{4}q^{n^{2}}. \] Further related estimates are due to K. BarrĂ© [J. Number Theory 66, 102-128 (1997; Zbl 0898.11030)] and P. Philippon [J. Number Theory 64, No. 2, 291-338 (1997; Zbl 0901.11026); Mesures d’approximation de valeurs de fonctions analytiques, submitted].
For the entire collection see [Zbl 0907.00013].

MSC:

11J91 Transcendence theory of other special functions
11J85 Algebraic independence; Gel’fond’s method
11J82 Measures of irrationality and of transcendence
11F11 Holomorphic modular forms of integral weight