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A uniqueness theorem for functions in the extended Selberg class

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We study the uniqueness of functions in the extended Selberg class. It was shown in Ki (Adv Math 231, 2484–2490, 2012) that if for a nonzero complex number \(c\) the inverse images \(L_1^{-1}(c)\) and \(L_2^{-1}(c)\) of two functions satisfying the same functional equation in the extended Selberg class are the same, then \(L_1(s)\) and \(L_2(s)\) are identical. Here we prove that this holds even without the assumption that they satisfy the same functional equation.

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Correspondence to Haseo Ki.

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The third named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002549).

Research of the first author was partially supported by NSF Grant DMS-1200582.

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Gonek, S.M., Haan, J. & Ki, H. A uniqueness theorem for functions in the extended Selberg class. Math. Z. 278, 995–1004 (2014).

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