On Berndt-Straub conjectures about zeros of generalized Ramanujan polynomials

We settle two conjectures of Berndt and Straub asserting that all nonzero zeros of the generalized Ramanujan polynomials $R_k(x;χ,ψ)$ and $S_{2n}(x;χ,χ)$ lie on the unit circle for real nonprincipal Dirichlet characters. We prove their conjecture for $R_k$. For $S_{2n}$, we prove that at most four n