The generalized Riemann hypothesis from zeros of a single \(L\)-function (Q6634417)

Let \(L(s,\chi)\) be the Dirichlet \(L\)-function attached to an arbitrary primitive character \(\chi\). For a fixed primitive character \(\chi\) modulo \(q\), we consider the following hypothesis about the zeros of \(L(s,\chi)\):\N\N\(GRH[\chi]\): if \(L(\beta+i\gamma,\chi)=0\) with \(\beta>0\), then \(\beta=1/2\). \N\NThe generalized Riemann Hypothesis \(GRH\): \(GRH[\chi]\) holds for all primitive character \(\chi\).\N\NIn this paper under review, the author proves mainly in Theorem 1.1 that for each primitive character \(\chi\), the \(GRH\) is equivalent to another hypothesis noted \(GRH^{\dagger}[\chi]\) which is the \(GRH[\chi]\) with further condition given in equation (1.3) page 1283. In fact, the author extends his previous result on the Riemann zeta function \(\zeta(s)\) to Dirichlet \(L\)-functions [\textit{W. Banks}, Res. Number Theory 10, No. 1, Paper No. 12, 9 p. (2024; Zbl 07801553)].