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2004 A remark on solutions of functional equations of Riemann's type
Jerzy Kaczorowski, Alberto Perelli
Funct. Approx. Comment. Math. 32: 51-55 (2004). DOI: 10.7169/facm/1538186624


It is proved that a general functional equation of the Riemann type with multiple gamma factors has non-trivial solutions in the space of generalized Dirichlet series. Moreover, for a fixed functional equation, the space of such solutions has uncountable basis. The proof is based on Hecke's theory of Dirichlet series associated with modular forms for the groups $G(\lambda)$. This is in constrast with the situation in the extended Selberg class where there exist functional equations without non-trivial solutions. Presumably this holds for non-integer degrees $d$, but up to date was confirmed only for $0 \leqslant d \lt 5/3$. In the case of $d = 0$ or $d = 1$ the space of solutions belonging to the extended Selberg class, through non-trivial, is finite dimensional.


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Jerzy Kaczorowski. Alberto Perelli. "A remark on solutions of functional equations of Riemann's type." Funct. Approx. Comment. Math. 32 51 - 55, 2004.


Published: 2004
First available in Project Euclid: 29 September 2018

zbMATH: 1065.11071
MathSciNet: MR2109943
Digital Object Identifier: 10.7169/facm/1538186624

Primary: 11M41

Keywords: Dirichlet series , Hecke modular forms , Riemann's functional equation , Selberg class

Rights: Copyright © 2004 Adam Mickiewicz University


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