Abstract
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.
Citation
Jerzy Kaczorowski. Alberto Perelli. "A remark on solutions of functional equations of Riemann's type." Funct. Approx. Comment. Math. 32 51 - 55, 2004. https://doi.org/10.7169/facm/1538186624
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