A remark on solutions of functional equations of Riemann's type
Let \(\mathcal C\) be a class of generalized Dirichlet series \[ F(s)=\sum_{n=1}^\infty\frac{a(n)}{l_n^s} \] absolutely convergent for \(\sigma\) sufficiently large, where \((l_n)\) is an increasing sequence of positive real numbers tending to \(\infty\), such that \(F(s)\) is an entire function of finite order. For given \(\omega\in\mathbb{C}\) with \(| \omega| =1\), \(Q>0\), \(\lambda=(\lambda_1, \dots, \lambda_r)\) with \(\lambda_j>0\) and \(\mu=(\mu_1,\dots, \mu_r)\) with \(\Re \mu_j\geq0\) the authors consider the functional equation \[ \Phi(s)=\omega\overline{\Phi(1-s)},\tag{\(*\)} \] where \(\overline{f}(s)=\overline{f(\overline{s})}\) and \[ \Phi(s)=Q^s\prod_{j=1}^r\Gamma(\lambda_j s+\mu_j) F(s). \] Denote by \(D(\lambda, \mu, Q, \omega)\) the real vector space of solutions \(F\in\mathcal C\) of functional equation \((*)\). Then the authors prove that \(D(\lambda, \mu, Q, \omega)\) has an uncountable basis. This result is in contrast with the situation in the extended Selberg class (special case of \(\mathcal C\), with \((l_n)=\mathbb{N}\)) where there exist functional equations without non-trivial solutions, see \textit{J. Kaczorowski} and \textit{A. Perelli} [Invent. Math. 150, No. 3, 485--516 (2002; Zbl 1034.11051)]. The proof is based on Hecke's theory of Dirichlet series associated with modular forms.
scientific article; zbMATH DE number 2149316