On the zeros of linear combinations of the Matsumoto zeta-functions
Let \(\phi_1(s),\dots, \phi_r(s)\) be Matsumoto zeta-functions, defined by generalized Euler products, and satisfying appropriate regularity conditions. Let \(V(s)\) be a non-trivial linear combination of \(\phi_1(s),\dots, \phi_r(s)\), and suppose that they are independent in a suitable sense. Then it is shown that \(V(s)\) has \(\gg T\) zeros in the region \(\sigma_1\leq\sigma\leq \sigma_2\), \(0\leq t\leq T\), for any \(\sigma_1<\sigma_2\) in the right-hand half of the critical strip. The proof is based on a multidimensional limit theorem for the distribution of the functions \(\phi_1(s),\dots, \phi_r(s)\), which produces regions in which \(V(s)\) approximates any desired analytic function.
scientific article; zbMATH DE number 1262940
scientific article