On the non-trivial zeros off the critical line for \(L\)-functions from the extended Selberg class
In the paper, the authors investigate the \(L\)-functions from the extended Selberg class \(S^\#\) [for the description of \(S^\#\), see, Acta Math. 182, 207--241 (1999; Zbl 1126.11335)]. Let \(N_F(\sigma,T)\) denote the number of zeros of \(F(s)\in S^\#\) lying in the region \(\Re s>1\), \(0<\Im s <T\). We say that the function \(F \in S^\#\) has the density property if \(N_F(\sigma, T)=o(T)\) as \(T \to \infty\), for every fixed \(\frac{1}{2} < \sigma<1\). Denote by \(S^\#_d\) the set of \(F \in S^\#_d\) with a given degree \(d\). Note that if \(F \in S^\#_1\) then \(N_F(\sigma, T)\ll T\), for every fixed \(\sigma>\frac{1}{2}\). The authors prove that a function \(F \in S^\#_1\) has the density if and only if \[ F(s+i\theta)=P(s)L(s,\chi) \] for certain real \(\theta\), a Dirichlet polynomial \(P\in S^\#_0\) and a Dirichlet \(L\)-function \(L(s,\chi)\) corresponding to a primitive Dirichlet character \(\chi\). Moreover, \(P\) satisfies the RH: \(P(s)\neq 0\), for \(\Re \neq \frac{1}{2}\). The second result deals with the function \[ F(s)=\sum_{j=1}^{N}P_j(s)L(s,\chi_j), \quad N \geq 2, \] where \(\chi_1,\dots,\chi_N\) are pairwise inequivalent Dirichlet characters, and \(P_1,\dots,P_N\) non-zero Dirichlet polynomials. It is proved that \(F(s)\) has infinitely many zeros off the critical line, i.e., for every \(\frac{1}{2} \leq \sigma_1<\sigma_2<1\), \[ N_F(\sigma_1,\sigma_2,T)\ll T, \quad T \to \infty, \] where \(N_F(\sigma_1,\sigma_2,T)=N_F(\sigma_1,T)-N_F(\sigma_2,T)\), \(\sigma_1<\sigma_2\). This result is a consequence of the universality property for Dirichlet \(L\)-functions.
scientific article; zbMATH DE number 5159770