A criterion for the generalized Riemann hypothesis (Q2655591)

We let \(L(s,f)\) be the automorphic \(L\)-function attached to a primitive holomorphic cusp form \(f\) and write, for \(\text{Re}\, s>1\), \(L(s,f)=\sum_{n\geq 1} \lambda_f(n)n^{-s}\). The author gives a criterion for the Generalized Riemann Hypothesis (GRH) for \(L(s,f)\). Define \[ \rho_{f,a}(x) =\sum_{n\leq 1/(ax)} \lambda_f(n). \] Then GRH for \(L(s,f)\) holds if and only if \(\chi_{(0,1]}\in \overline{S_f^{nat}}\), where \(\chi_{(0,1]}\) is the characteristic function of the interval \((0,1]\) and \(S_f^{nat}\) is the linear null of \(\{\rho_{f,a}: \, a=1,2,\cdots\}\). The bar denotes the closure in the \(L^2\) real-valued function space with respect to the inner product \(\langle f,g\rangle = \int_0^\infty f(x)g(x)\, dx\) (and the measure is the Euclidean measure). The method of proof is based on the idea of \textit{L. Báez-Duarte} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 14, No. 1, 5--11 (2003; Zbl 1097.11041)].