Abstract
We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF p. This implies that the equationF a=Gb with (a, b)=1 has the unique solutionF=H b andG=H a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.
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Molteni, G. On the algebraic independence in the selberg class. Arch. Math 79, 432–438 (2002). https://doi.org/10.1007/BF02638380
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DOI: https://doi.org/10.1007/BF02638380