We prove that a function F of the Selberg class script L sign is a b-th power in script capital L sign, i.e., F = Hb for some H ∈ script L sign, if and only if b divides the order of every zero of F and of every p-component Fp. This implies that the equation Fa = Gb with (a, b) = 1 has the unique solution F = Hb and G = Ha in script L sign. As a consequence, we prove that if F and G are distinct primitive elements of script L sign, then the transcendence degree of ℂ[F, G] over ℂ is two.
On the algebraic independence in the Selberg class / G. Molteni. - In: ARCHIV DER MATHEMATIK. - ISSN 0003-889X. - 79:6(2002 Dec), pp. 432-438. [10.1007/BF02638380]
On the algebraic independence in the Selberg class
G. Molteni
2002
Abstract
We prove that a function F of the Selberg class script L sign is a b-th power in script capital L sign, i.e., F = Hb for some H ∈ script L sign, if and only if b divides the order of every zero of F and of every p-component Fp. This implies that the equation Fa = Gb with (a, b) = 1 has the unique solution F = Hb and G = Ha in script L sign. As a consequence, we prove that if F and G are distinct primitive elements of script L sign, then the transcendence degree of ℂ[F, G] over ℂ is two.
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