The four exponentials conjecture and D. Bertrand’s conjecture on the modular function. (La conjecture des quatre exponentielles et les conjectures de D. Bertrand sur la fonction modulaire.) (French) Zbl 0887.11030

The four exponentials conjecture has been proposed first by Th. Schneider in 1957 (problem 1 of his book [Einführung in die transzendenten Zahlen. Berlin etc.: Springer-Verlag (1957; Zbl 0077.04703)]), then by S. Lang [Introduction to transcendental numbers. Reading, Mass. etc.: Addison-Wesley (1966; Zbl 0144.04101)] and by K. Ramachandra [Contributions to the theory of transcendental numbers. I, II. Acta Arith. 14, 65–72, 73–88 (1968; Zbl 0176.33101)]: Let \((x_1,x_2)\) and \((y_1,y_2)\) be two pairs of \(\mathbb{Q}\)-linearly independent complex numbers. Then at least one of the four numbers \(e^{x_1y_1}\), \(e^{x_1y_2}\), \(e^{x_2y_1}\), \(e^{x_2y_2}\) is transcendental.
D. Bertrand [Theta functions and transcendence, Ramanujan J. 1, No. 4, 339–350 (1997; Zbl 0916.11043)] suggested several conjectures related with the modular function \(J\), in particular the following: Let \(q_1\) and \(q_2\) be two multiplicatively independent algebraic numbers in the domain \(0<|q|<1\). Then \(J(q_1)\) and \(J(q_2)\) are algebraically independent. He also pointed out that this problem would solve the following special case of the four exponentials conjecture (with \(x_1=1\), \(x_2=(\log\alpha_1)/2i\pi\), \(y_1 =2i\pi\), \(y_2 =\log\alpha_2\)): If \(\alpha_1\) and \(\alpha_2\) are positive real numbers \(\ne 1\), then \((\log\alpha_1)(\log\alpha_2)/\pi^{2}\) is irrational.
The author introduces further conjectures and gives a careful analysis of their relationship. For instance he shows that Bertrand’s above mentioned conjecture is equivalent to \(6\) other statements, one of them being: For any \(\tau\) in the upper half plane, at least one of the two numbers \(e^{2i\pi\tau}\) and \(e^{-2i\pi/\tau}\) is transcendental. He shows that this statement holds in the following 5 cases:
(i) \(\tau\) is algebraic over the field \(\mathbb{Q}(\pi)\),
(ii) The real part \(\operatorname{Re} \tau\) of \(\tau\) is algebraic \(\ne 0\),
(iii) The imaginary part \(\operatorname{Im} \tau\) of \(\tau\) is algebraic,
(iv) \((\operatorname{Re} \tau)/|\tau|^2\) is algebraic \(\ne 0\),
(v) \((\operatorname{Im} \tau)/|\tau|^2\) is algebraic.
The author also deduces from the four exponentials conjecture the following statement: For any \(z\in\mathbb{C}\) with \(|z|=1\) and \(z\not=\pm 1\), the number \(e^{2i\pi z}\) is transcendental.
From these connections between modular functions and the exponential function, one can expect further progress on either side.


11J81 Transcendence (general theory)
11J91 Transcendence theory of other special functions
11F03 Modular and automorphic functions
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