Is the first zero 2.40482555769577... of the Bessel function $J_0(x)$ transcendental? (Is anything known about its status?)
2026-03-25 23:38:19.1774481899
Rationality and transcendence of zeros of Bessel functions
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Yes, it is transcendental since for all non zero $\alpha\in\overline{\mathbb Q}$, $J_0(\alpha)$ is transcendental. It is a result of Siegel (1929)
Original Siegel Paper Uber einige Anwendungen diophantischer Approximationen