Selberg's method extended to degree two: a proof that a positive proportion of the non-trivial zeros of a linear combination of degree-two L-functions (concretely, Hecke L-functions with complex ideal-class-group characters of an imaginary quadratic field, attached to automorphic cusp forms) lie on the critical line, even though such a linear combination has no Euler product and hence violates the Riemann Hypothesis. The argument unifies Selberg's positive-proportion theorem with his density theorem for each constituent L-function, driven by mollified mean-value estimates and a value-distribution result, and (under Selberg's orthogonality conjecture) applies to general linear combinations satisfying a Riemann-type functional equation.
A math.NT research article (I.S. Rezvyakova, published in Izvestiya: Mathematics 2016) giving a detailed, self-contained proof, by Selberg's method, that a positive proportion of the non-trivial zero…