Unique factorization results for semigroups of \(L\)-functions
The extended Selberg class \(S^{\#}\) is defined as the class of non-identically vanishing Dirichlet series \[ F(s)=\sum_{n=1}^{\infty}a_F(n)n^{-s},\quad s=\sigma+it, \] absolutely convergent for \(\sigma>1\). The function \(F(s)\) satisfies a functional equation \[ \Phi(s)=\omega {\overline {\Phi(\overline{1-s})}}, \] where \(|\omega|=1\) and \[ \Phi(s)=Q^s\prod_{j=1}^{r}\Gamma(\lambda_j s+\mu_j)F(s) \] with \(r \geq 0\), \(Q>0\), \(\lambda_j>0\) and \(\Re \mu_j \geq 0\) (\(r=0\) meaning that there are no \(\Gamma\)-factors). The degree of a function \(F \in S^{\#}\) is defined by \[ d_F=2\sum_{j=1}^{r}\lambda_j. \] In the paper, the authors prove that the set of functions \(F \in S^{\#}\) with degree 0 has unique factoriation. Also, they consider the multiplicative semigroups generated by the functions \(F \in S^{\#}\) with degree 1, and prove that such set has unique factorization. The authors consider some extensions of above mentioned results leading to similar results involving certain classical \(L\)-functions.
scientific article; zbMATH DE number 5284754
scientific article