On the distribution of zeros of linear combinations of Euler products
Let \(L_j (s)\) be functions defined by Dirichlet series, possessing Euler products, and all satisfying the same functional equation. The purpose of the present paper is to consider zeros of the linear combination \(f(s)= \sum^N_1 \lambda_j L_j (s)\). Let \(N(T)\) be the number of zeros of \(f\) up to height \(T\), and let \(N_0 (T)\) be the number of such zeros which are simple and lie on the critical line. Subject to suitable hypotheses it is then shown that both \(N(T)\) and \(N_0 (T)\) are asymptotic to \(CT \log T\), where \(C\) is a constant arising from the form of the \(\Gamma\)-factors in the functional equation. The proof is quite difficult, but its essence is to show that the behaviour of \(f(s)\) is usually dominated by just one of the functions \(L_j (s)\).
scientific article; zbMATH DE number 871846
scientific article